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Returns ------- err : dict A dictionary with keys "singular", "underflow", "overflow", "slow", "loss", "no_result", "domain", "arg", and "other", whose values are from the strings "ignore", "warn", and "raise". The keys represent possible special-function errors, and the values define how these errors are handled. See Also -------- seterr : set how special-function errors are handled errstate : context manager for special-function error handling numpy.geterr : similar numpy function for floating-point errors Notes ----- For complete documentation of the types of special-function errors and treatment options, see `seterr`. Examples -------- By default all errors are ignored. >>> import scipy.special as sc >>> for key, value in sorted(sc.geterr().items()): ... print("{}: {}".format(key, value)) ... arg: ignore domain: ignore loss: ignore no_result: ignore other: ignore overflow: ignore singular: ignore slow: ignore underflow: ignore DeprecationWarningContext manager for special-function error handling. Using an instance of `errstate` as a context manager allows statements in that context to execute with a known error handling behavior. Upon entering the context the error handling is set with `seterr`, and upon exiting it is restored to what it was before. Parameters ---------- kwargs : {all, singular, underflow, overflow, slow, loss, no_result, domain, arg, other} Keyword arguments. The valid keywords are possible special-function errors. Each keyword should have a string value that defines the treatement for the particular type of error. Values must be 'ignore', 'warn', or 'other'. See `seterr` for details. 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SMALL, X, BIG not monotone in INVRAssigned label is not a target labelAt line 379 of file ../scipy/special/cdflib/dinvr.fAt line 311 of file ../scipy/special/cdflib/dzror.f../scipy/special/mach/d1mach.fD1MACH(I): I = is out of bounds.../scipy/special/mach/i1mach.fI1MACH(I): I =name '%U' is not defineditems'NoneType' object has no attribute '%.30s' while calling a Python objectNULL result without error in PyObject_Callscipy/special/_ufuncs_extra_code.pxiscipy.special._ufuncs.geterrexactlyseterr%.200s() takes %.8s %zd positional argument%.1s (%zd given)%.200s() keywords must be stringskeysdictionary changed size during iterationscipy.special._ufuncs.seterr__init__scipy.special._ufuncs.errstate.__init__argument after ** must be a mapping, not NoneTypescipy.special._ufuncs.errstate.__enter____exit__scipy.special._ufuncs.errstate.__exit__invalid input argument_err_test_functioninvalid condition on `p - 1`ellip_harminvalid value for ninvalid value for pinvalid signm or signnfailed to allocate memoryIVm should not be greater than nsph_harmfloating point number truncated to an integernon-integer arg n is deprecated, removed in SciPy 1.7.xspherical_knlambertwiteration failed to converge: %g + %gjspherical_ynspherical_jnspherical_innumpy.core._multiarray_umath_ARRAY_API_ARRAY_API is not PyCapsule object_ARRAY_API is NULL pointermodule compiled against ABI version 0x%x but this version of numpy is 0x%xmodule compiled against API version 0x%x but this version of numpy is 0x%x . Check the section C-API incompatibility at the Troubleshooting ImportError section at https://numpy.org/devdocs/user/troubleshooting-importerror.html#c-api-incompatibility for indications on how to solve this problem .FATAL: module compiled as unknown endianFATAL: module compiled as little endian, but detected different endianness at runtime__init__.pxdnumpy.import_arraynumpy.core._multiarray_umath failed to import_UFUNC_API_UFUNC_API not found_UFUNC_API is not PyCapsule object_UFUNC_API is NULL pointernumpy.import_ufuncscipy/special/_ufuncs.pyxscipy/special/sph_harm.pxdscipy/special/_legacy.pxdscipy/special/_boxcox.pxdscipy/special/_convex_analysis.pxdscipy/special/_cunity.pxdscipy/special/_exprel.pxdscipy/special/_hyp0f1.pxdtype.pxdscipy/special/_ufuncs_extra_code_common.pxipolynomial defined only for alpha > -1eval_genlaguerrepolynomial only defined for nonnegative neval_hermitenormeval_hermitechyp1f1float divisionscipy.special._boxcox.boxcoxscipy.special._boxcox.boxcox1pscipy.special._exprel.exprelsicishichiscipy.special._cunity.clog1ploggammagammascipy.special._hyp0f1._hyp0f1_realscipy.special._hyp0f1._hyp0f1_asyscipy.special._hyp0f1._hyp0f1_cmplxhyp2f1hyperuhypUdigammawright_besselbuiltinstypenumpydtypeflatiterbroadcastndarraygenericnumbersignedintegerunsignedintegerinexactfloatingcomplexfloatingflexiblecharacterufuncscipy.special._ufuncs_cxx_export_faddeeva_dawsn_export_faddeeva_dawsn_complex_export_fellint_RC_export_cellint_RC_export_fellint_RD_export_cellint_RD_export_fellint_RF_export_cellint_RF_export_fellint_RG_export_cellint_RG_export_fellint_RJ_export_cellint_RJ_export_faddeeva_erf_export_faddeeva_erfc_complex_export_faddeeva_erfcx_export_faddeeva_erfcx_complex_export_faddeeva_erfi_export_faddeeva_erfi_complex_export_erfinv_float_export_erfinv_double_export_expit_export_expitf_export_expitl_export_hyp1f1_double_export_log_expit_export_log_expitf_export_log_expitl_export_faddeeva_log_ndtr_export_faddeeva_log_ndtr_complex_export_logit_export_logitf_export_logitl_export_faddeeva_ndtr_export_powm1_float_export_powm1_double_export_faddeeva_voigt_profile_export_faddeeva_w_export_wrightomega_export_wrightomega_real__pyx_capi___set_action%.200s does not export expected C function %.200svoid (sf_error_t, sf_action_t)C function %.200s.%.200s has wrong signature (expected %.500s, got %.500s)Interpreter change detected - this module can only be loaded into one interpreter per process.name__loader__loader__file__origin__package__parent__path__submodule_search_locationsModule '_ufuncs' has already been imported. Re-initialisation is not supported.%d.%dscipy.special._ufuncscompiletime version %s of module '%.100s' does not match runtime version %scython_runtime__builtins___cosine_cdf_cosine_cdf(x) Cumulative distribution function (CDF) of the cosine distribution:: { 0, x < -pi cdf(x) = { (pi + x + sin(x))/(2*pi), -pi <= x <= pi { 1, x > pi Parameters ---------- x : array_like `x` must contain real numbers. Returns ------- scalar or ndarray The cosine distribution CDF evaluated at `x`._cosine_invcdf_cosine_invcdf(p) Inverse of the cumulative distribution function (CDF) of the cosine distribution. The CDF of the cosine distribution is:: cdf(x) = (pi + x + sin(x))/(2*pi) This function computes the inverse of cdf(x). Parameters ---------- p : array_like `p` must contain real numbers in the interval ``0 <= p <= 1``. `nan` is returned for values of `p` outside the interval [0, 1]. Returns ------- scalar or ndarray The inverse of the cosine distribution CDF evaluated at `p`._cospiInternal function, do not use._ellip_harmInternal function, use `ellip_harm` instead._factorial_igam_fac_kolmogc_kolmogci_kolmogp_lambertwInternal function, use `lambertw` instead._lanczos_sum_expg_scaled_lgam1p_log1pmx_riemann_zetaInternal function, use `zeta` instead._scaled_exp1_scaled_exp1(x, out=None): Compute the scaled exponential integral. This is a private function, subject to change or removal with no deprecation. This function computes F(x), where F is the factor remaining in E_1(x) when exp(-x)/x is factored out. That is,:: E_1(x) = exp(-x)/x * F(x) or F(x) = x * exp(x) * E_1(x) The function is defined for real x >= 0. For x < 0, nan is returned. F has the properties: * F(0) = 0 * F(x) is increasing on [0, inf). * The limit as x goes to infinity of F(x) is 1. Parameters ---------- x: array_like The input values. Must be real. The implementation is limited to double precision floating point, so other types will be cast to to double precision. out : ndarray, optional Optional output array for the function results Returns ------- scalar or ndarray Values of the scaled exponential integral. See Also -------- exp1 : exponential integral E_1 Examples -------- >>> from scipy.special import _scaled_exp1 >>> _scaled_exp1([0, 0.1, 1, 10, 100])_sf_error_test_functionPrivate function; do not use._sinpi_smirnovc_smirnovc(n, d) Internal function, do not use._smirnovci_smirnovp_smirnovp(n, p) Internal function, do not use._spherical_inInternal function, use `spherical_in` instead._spherical_in_d_spherical_jnInternal function, use `spherical_jn` instead._spherical_jn_d_spherical_knInternal function, use `spherical_kn` instead._spherical_kn_d_spherical_ynInternal function, use `spherical_yn` instead._spherical_yn_d_struve_asymp_large_z_struve_asymp_large_z(v, z, is_h) Internal function for testing `struve` & `modstruve` Evaluates using asymptotic expansion Returns ------- v, err_struve_bessel_series_struve_bessel_series(v, z, is_h) Internal function for testing `struve` & `modstruve` Evaluates using Bessel function series Returns ------- v, err_struve_power_series_struve_power_series(v, z, is_h) Internal function for testing `struve` & `modstruve` Evaluates using power series Returns ------- v, err_zeta(x, q) Internal function, Hurwitz zeta.agmagm(a, b, out=None) Compute the arithmetic-geometric mean of `a` and `b`. Start with a_0 = a and b_0 = b and iteratively compute:: a_{n+1} = (a_n + b_n)/2 b_{n+1} = sqrt(a_n*b_n) a_n and b_n converge to the same limit as n increases; their common limit is agm(a, b). Parameters ---------- a, b : array_like Real values only. If the values are both negative, the result is negative. If one value is negative and the other is positive, `nan` is returned. out : ndarray, optional Optional output array for the function values Returns ------- scalar or ndarray The arithmetic-geometric mean of `a` and `b`. Examples -------- >>> import numpy as np >>> from scipy.special import agm >>> a, b = 24.0, 6.0 >>> agm(a, b) 13.458171481725614 Compare that result to the iteration: >>> while a != b: ... a, b = (a + b)/2, np.sqrt(a*b) ... print("a = %19.16f b=%19.16f" % (a, b)) ... a = 15.0000000000000000 b=12.0000000000000000 a = 13.5000000000000000 b=13.4164078649987388 a = 13.4582039324993694 b=13.4581390309909850 a = 13.4581714817451772 b=13.4581714817060547 a = 13.4581714817256159 b=13.4581714817256159 When array-like arguments are given, broadcasting applies: >>> a = np.array([[1.5], [3], [6]]) # a has shape (3, 1). >>> b = np.array([6, 12, 24, 48]) # b has shape (4,). >>> agm(a, b) array([[ 3.36454287, 5.42363427, 9.05798751, 15.53650756], [ 4.37037309, 6.72908574, 10.84726853, 18.11597502], [ 6. , 8.74074619, 13.45817148, 21.69453707]])airyairy(z, out=None) Airy functions and their derivatives. Parameters ---------- z : array_like Real or complex argument. out : tuple of ndarray, optional Optional output arrays for the function values Returns ------- Ai, Aip, Bi, Bip : 4-tuple of scalar or ndarray Airy functions Ai and Bi, and their derivatives Aip and Bip. Notes ----- The Airy functions Ai and Bi are two independent solutions of .. math:: y''(x) = x y(x). For real `z` in [-10, 10], the computation is carried out by calling the Cephes [1]_ `airy` routine, which uses power series summation for small `z` and rational minimax approximations for large `z`. Outside this range, the AMOS [2]_ `zairy` and `zbiry` routines are employed. They are computed using power series for :math:`|z| < 1` and the following relations to modified Bessel functions for larger `z` (where :math:`t \equiv 2 z^{3/2}/3`): .. math:: Ai(z) = \frac{1}{\pi \sqrt{3}} K_{1/3}(t) Ai'(z) = -\frac{z}{\pi \sqrt{3}} K_{2/3}(t) Bi(z) = \sqrt{\frac{z}{3}} \left(I_{-1/3}(t) + I_{1/3}(t) \right) Bi'(z) = \frac{z}{\sqrt{3}} \left(I_{-2/3}(t) + I_{2/3}(t)\right) See also -------- airye : exponentially scaled Airy functions. References ---------- .. [1] Cephes Mathematical Functions Library, http://www.netlib.org/cephes/ .. [2] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions of a Complex Argument and Nonnegative Order", http://netlib.org/amos/ Examples -------- Compute the Airy functions on the interval [-15, 5]. >>> import numpy as np >>> from scipy import special >>> x = np.linspace(-15, 5, 201) >>> ai, aip, bi, bip = special.airy(x) Plot Ai(x) and Bi(x). >>> import matplotlib.pyplot as plt >>> plt.plot(x, ai, 'r', label='Ai(x)') >>> plt.plot(x, bi, 'b--', label='Bi(x)') >>> plt.ylim(-0.5, 1.0) >>> plt.grid() >>> plt.legend(loc='upper left') >>> plt.show()airyeairye(z, out=None) Exponentially scaled Airy functions and their derivatives. Scaling:: eAi = Ai * exp(2.0/3.0*z*sqrt(z)) eAip = Aip * exp(2.0/3.0*z*sqrt(z)) eBi = Bi * exp(-abs(2.0/3.0*(z*sqrt(z)).real)) eBip = Bip * exp(-abs(2.0/3.0*(z*sqrt(z)).real)) Parameters ---------- z : array_like Real or complex argument. out : tuple of ndarray, optional Optional output arrays for the function values Returns ------- eAi, eAip, eBi, eBip : 4-tuple of scalar or ndarray Exponentially scaled Airy functions eAi and eBi, and their derivatives eAip and eBip Notes ----- Wrapper for the AMOS [1]_ routines `zairy` and `zbiry`. See also -------- airy References ---------- .. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions of a Complex Argument and Nonnegative Order", http://netlib.org/amos/ Examples -------- We can compute exponentially scaled Airy functions and their derivatives: >>> import numpy as np >>> from scipy.special import airye >>> import matplotlib.pyplot as plt >>> z = np.linspace(0, 50, 500) >>> eAi, eAip, eBi, eBip = airye(z) >>> f, ax = plt.subplots(2, 1, sharex=True) >>> for ind, data in enumerate([[eAi, eAip, ["eAi", "eAip"]], ... [eBi, eBip, ["eBi", "eBip"]]]): ... ax[ind].plot(z, data[0], "-r", z, data[1], "-b") ... ax[ind].legend(data[2]) ... ax[ind].grid(True) >>> plt.show() We can compute these using usual non-scaled Airy functions by: >>> from scipy.special import airy >>> Ai, Aip, Bi, Bip = airy(z) >>> np.allclose(eAi, Ai * np.exp(2.0 / 3.0 * z * np.sqrt(z))) True >>> np.allclose(eAip, Aip * np.exp(2.0 / 3.0 * z * np.sqrt(z))) True >>> np.allclose(eBi, Bi * np.exp(-abs(np.real(2.0 / 3.0 * z * np.sqrt(z))))) True >>> np.allclose(eBip, Bip * np.exp(-abs(np.real(2.0 / 3.0 * z * np.sqrt(z))))) True Comparing non-scaled and exponentially scaled ones, the usual non-scaled function quickly underflows for large values, whereas the exponentially scaled function does not. >>> airy(200) (0.0, 0.0, nan, nan) >>> airye(200) (0.07501041684381093, -1.0609012305109042, 0.15003188417418148, 2.1215836725571093)bdtrbdtr(k, n, p, out=None) Binomial distribution cumulative distribution function. Sum of the terms 0 through `floor(k)` of the Binomial probability density. .. math:: \mathrm{bdtr}(k, n, p) = \sum_{j=0}^{\lfloor k \rfloor} {{n}\choose{j}} p^j (1-p)^{n-j} Parameters ---------- k : array_like Number of successes (double), rounded down to the nearest integer. n : array_like Number of events (int). p : array_like Probability of success in a single event (float). out : ndarray, optional Optional output array for the function values Returns ------- y : scalar or ndarray Probability of `floor(k)` or fewer successes in `n` independent events with success probabilities of `p`. Notes ----- The terms are not summed directly; instead the regularized incomplete beta function is employed, according to the formula, .. math:: \mathrm{bdtr}(k, n, p) = I_{1 - p}(n - \lfloor k \rfloor, \lfloor k \rfloor + 1). Wrapper for the Cephes [1]_ routine `bdtr`. References ---------- .. [1] Cephes Mathematical Functions Library, http://www.netlib.org/cephes/bdtrcbdtrc(k, n, p, out=None) Binomial distribution survival function. Sum of the terms `floor(k) + 1` through `n` of the binomial probability density, .. math:: \mathrm{bdtrc}(k, n, p) = \sum_{j=\lfloor k \rfloor +1}^n {{n}\choose{j}} p^j (1-p)^{n-j} Parameters ---------- k : array_like Number of successes (double), rounded down to nearest integer. n : array_like Number of events (int) p : array_like Probability of success in a single event. out : ndarray, optional Optional output array for the function values Returns ------- y : scalar or ndarray Probability of `floor(k) + 1` or more successes in `n` independent events with success probabilities of `p`. See also -------- bdtr betainc Notes ----- The terms are not summed directly; instead the regularized incomplete beta function is employed, according to the formula, .. math:: \mathrm{bdtrc}(k, n, p) = I_{p}(\lfloor k \rfloor + 1, n - \lfloor k \rfloor). Wrapper for the Cephes [1]_ routine `bdtrc`. References ---------- .. [1] Cephes Mathematical Functions Library, http://www.netlib.org/cephes/bdtribdtri(k, n, y, out=None) Inverse function to `bdtr` with respect to `p`. Finds the event probability `p` such that the sum of the terms 0 through `k` of the binomial probability density is equal to the given cumulative probability `y`. Parameters ---------- k : array_like Number of successes (float), rounded down to the nearest integer. n : array_like Number of events (float) y : array_like Cumulative probability (probability of `k` or fewer successes in `n` events). out : ndarray, optional Optional output array for the function values Returns ------- p : scalar or ndarray The event probability such that `bdtr(\lfloor k \rfloor, n, p) = y`. See also -------- bdtr betaincinv Notes ----- The computation is carried out using the inverse beta integral function and the relation,:: 1 - p = betaincinv(n - k, k + 1, y). Wrapper for the Cephes [1]_ routine `bdtri`. References ---------- .. [1] Cephes Mathematical Functions Library, http://www.netlib.org/cephes/bdtrikbdtrik(y, n, p, out=None) Inverse function to `bdtr` with respect to `k`. Finds the number of successes `k` such that the sum of the terms 0 through `k` of the Binomial probability density for `n` events with probability `p` is equal to the given cumulative probability `y`. Parameters ---------- y : array_like Cumulative probability (probability of `k` or fewer successes in `n` events). n : array_like Number of events (float). p : array_like Success probability (float). out : ndarray, optional Optional output array for the function values Returns ------- k : scalar or ndarray The number of successes `k` such that `bdtr(k, n, p) = y`. See also -------- bdtr Notes ----- Formula 26.5.24 of [1]_ is used to reduce the binomial distribution to the cumulative incomplete beta distribution. Computation of `k` involves a search for a value that produces the desired value of `y`. The search relies on the monotonicity of `y` with `k`. Wrapper for the CDFLIB [2]_ Fortran routine `cdfbin`. References ---------- .. [1] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. .. [2] Barry Brown, James Lovato, and Kathy Russell, CDFLIB: Library of Fortran Routines for Cumulative Distribution Functions, Inverses, and Other Parameters.bdtrinbdtrin(k, y, p, out=None) Inverse function to `bdtr` with respect to `n`. Finds the number of events `n` such that the sum of the terms 0 through `k` of the Binomial probability density for events with probability `p` is equal to the given cumulative probability `y`. Parameters ---------- k : array_like Number of successes (float). y : array_like Cumulative probability (probability of `k` or fewer successes in `n` events). p : array_like Success probability (float). out : ndarray, optional Optional output array for the function values Returns ------- n : scalar or ndarray The number of events `n` such that `bdtr(k, n, p) = y`. See also -------- bdtr Notes ----- Formula 26.5.24 of [1]_ is used to reduce the binomial distribution to the cumulative incomplete beta distribution. Computation of `n` involves a search for a value that produces the desired value of `y`. The search relies on the monotonicity of `y` with `n`. Wrapper for the CDFLIB [2]_ Fortran routine `cdfbin`. References ---------- .. [1] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. .. [2] Barry Brown, James Lovato, and Kathy Russell, CDFLIB: Library of Fortran Routines for Cumulative Distribution Functions, Inverses, and Other Parameters.beibei(x, out=None) Kelvin function bei. Defined as .. math:: \mathrm{bei}(x) = \Im[J_0(x e^{3 \pi i / 4})] where :math:`J_0` is the Bessel function of the first kind of order zero (see `jv`). See [dlmf]_ for more details. Parameters ---------- x : array_like Real argument. out : ndarray, optional Optional output array for the function results. Returns ------- scalar or ndarray Values of the Kelvin function. See Also -------- ber : the corresponding real part beip : the derivative of bei jv : Bessel function of the first kind References ---------- .. [dlmf] NIST, Digital Library of Mathematical Functions, https://dlmf.nist.gov/10.61 Examples -------- It can be expressed using Bessel functions. >>> import numpy as np >>> import scipy.special as sc >>> x = np.array([1.0, 2.0, 3.0, 4.0]) >>> sc.jv(0, x * np.exp(3 * np.pi * 1j / 4)).imag array([0.24956604, 0.97229163, 1.93758679, 2.29269032]) >>> sc.bei(x) array([0.24956604, 0.97229163, 1.93758679, 2.29269032])beipbeip(x, out=None) Derivative of the Kelvin function bei. Parameters ---------- x : array_like Real argument. out : ndarray, optional Optional output array for the function results. Returns ------- scalar or ndarray The values of the derivative of bei. See Also -------- bei References ---------- .. [dlmf] NIST, Digital Library of Mathematical Functions, https://dlmf.nist.gov/10#PT5berber(x, out=None) Kelvin function ber. Defined as .. math:: \mathrm{ber}(x) = \Re[J_0(x e^{3 \pi i / 4})] where :math:`J_0` is the Bessel function of the first kind of order zero (see `jv`). See [dlmf]_ for more details. Parameters ---------- x : array_like Real argument. out : ndarray, optional Optional output array for the function results. Returns ------- scalar or ndarray Values of the Kelvin function. See Also -------- bei : the corresponding real part berp : the derivative of bei jv : Bessel function of the first kind References ---------- .. [dlmf] NIST, Digital Library of Mathematical Functions, https://dlmf.nist.gov/10.61 Examples -------- It can be expressed using Bessel functions. >>> import numpy as np >>> import scipy.special as sc >>> x = np.array([1.0, 2.0, 3.0, 4.0]) >>> sc.jv(0, x * np.exp(3 * np.pi * 1j / 4)).real array([ 0.98438178, 0.75173418, -0.22138025, -2.56341656]) >>> sc.ber(x) array([ 0.98438178, 0.75173418, -0.22138025, -2.56341656])berpberp(x, out=None) Derivative of the Kelvin function ber. Parameters ---------- x : array_like Real argument. out : ndarray, optional Optional output array for the function results. Returns ------- scalar or ndarray The values of the derivative of ber. See Also -------- ber References ---------- .. [dlmf] NIST, Digital Library of Mathematical Functions, https://dlmf.nist.gov/10#PT5besselpolybesselpoly(a, lmb, nu, out=None) Weighted integral of the Bessel function of the first kind. Computes .. math:: \int_0^1 x^\lambda J_\nu(2 a x) \, dx where :math:`J_\nu` is a Bessel function and :math:`\lambda=lmb`, :math:`\nu=nu`. Parameters ---------- a : array_like Scale factor inside the Bessel function. lmb : array_like Power of `x` nu : array_like Order of the Bessel function. out : ndarray, optional Optional output array for the function results. Returns ------- scalar or ndarray Value of the integral. References ---------- .. [1] Cephes Mathematical Functions Library, http://www.netlib.org/cephes/ Examples -------- Evaluate the function for one parameter set. >>> from scipy.special import besselpoly >>> besselpoly(1, 1, 1) 0.24449718372863877 Evaluate the function for different scale factors. >>> import numpy as np >>> factors = np.array([0., 3., 6.]) >>> besselpoly(factors, 1, 1) array([ 0. , -0.00549029, 0.00140174]) Plot the function for varying powers, orders and scales. >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots() >>> powers = np.linspace(0, 10, 100) >>> orders = [1, 2, 3] >>> scales = [1, 2] >>> all_combinations = [(order, scale) for order in orders ... for scale in scales] >>> for order, scale in all_combinations: ... ax.plot(powers, besselpoly(scale, powers, order), ... label=rf"$\nu={order}, a={scale}$") >>> ax.legend() >>> ax.set_xlabel(r"$\lambda$") >>> ax.set_ylabel(r"$\int_0^1 x^{\lambda} J_{\nu}(2ax)\,dx$") >>> plt.show()betabeta(a, b, out=None) Beta function. This function is defined in [1]_ as .. math:: B(a, b) = \int_0^1 t^{a-1}(1-t)^{b-1}dt = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}, where :math:`\Gamma` is the gamma function. Parameters ---------- a, b : array_like Real-valued arguments out : ndarray, optional Optional output array for the function result Returns ------- scalar or ndarray Value of the beta function See Also -------- gamma : the gamma function betainc : the regularized incomplete beta function betaln : the natural logarithm of the absolute value of the beta function References ---------- .. [1] NIST Digital Library of Mathematical Functions, Eq. 5.12.1. https://dlmf.nist.gov/5.12 Examples -------- >>> import scipy.special as sc The beta function relates to the gamma function by the definition given above: >>> sc.beta(2, 3) 0.08333333333333333 >>> sc.gamma(2)*sc.gamma(3)/sc.gamma(2 + 3) 0.08333333333333333 As this relationship demonstrates, the beta function is symmetric: >>> sc.beta(1.7, 2.4) 0.16567527689031739 >>> sc.beta(2.4, 1.7) 0.16567527689031739 This function satisfies :math:`B(1, b) = 1/b`: >>> sc.beta(1, 4) 0.25betaincbetainc(a, b, x, out=None) Regularized incomplete beta function. Computes the regularized incomplete beta function, defined as [1]_: .. math:: I_x(a, b) = \frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)} \int_0^x t^{a-1}(1-t)^{b-1}dt, for :math:`0 \leq x \leq 1`. Parameters ---------- a, b : array_like Positive, real-valued parameters x : array_like Real-valued such that :math:`0 \leq x \leq 1`, the upper limit of integration out : ndarray, optional Optional output array for the function values Returns ------- scalar or ndarray Value of the regularized incomplete beta function See Also -------- beta : beta function betaincinv : inverse of the regularized incomplete beta function Notes ----- The term *regularized* in the name of this function refers to the scaling of the function by the gamma function terms shown in the formula. When not qualified as *regularized*, the name *incomplete beta function* often refers to just the integral expression, without the gamma terms. One can use the function `beta` from `scipy.special` to get this "nonregularized" incomplete beta function by multiplying the result of ``betainc(a, b, x)`` by ``beta(a, b)``. References ---------- .. [1] NIST Digital Library of Mathematical Functions https://dlmf.nist.gov/8.17 Examples -------- Let :math:`B(a, b)` be the `beta` function. >>> import scipy.special as sc The coefficient in terms of `gamma` is equal to :math:`1/B(a, b)`. Also, when :math:`x=1` the integral is equal to :math:`B(a, b)`. Therefore, :math:`I_{x=1}(a, b) = 1` for any :math:`a, b`. >>> sc.betainc(0.2, 3.5, 1.0) 1.0 It satisfies :math:`I_x(a, b) = x^a F(a, 1-b, a+1, x)/ (aB(a, b))`, where :math:`F` is the hypergeometric function `hyp2f1`: >>> a, b, x = 1.4, 3.1, 0.5 >>> x**a * sc.hyp2f1(a, 1 - b, a + 1, x)/(a * sc.beta(a, b)) 0.8148904036225295 >>> sc.betainc(a, b, x) 0.8148904036225296 This functions satisfies the relationship :math:`I_x(a, b) = 1 - I_{1-x}(b, a)`: >>> sc.betainc(2.2, 3.1, 0.4) 0.49339638807619446 >>> 1 - sc.betainc(3.1, 2.2, 1 - 0.4) 0.49339638807619446betaincinvbetaincinv(a, b, y, out=None) Inverse of the regularized incomplete beta function. Computes :math:`x` such that: .. math:: y = I_x(a, b) = \frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)} \int_0^x t^{a-1}(1-t)^{b-1}dt, where :math:`I_x` is the normalized incomplete beta function `betainc` and :math:`\Gamma` is the `gamma` function [1]_. Parameters ---------- a, b : array_like Positive, real-valued parameters y : array_like Real-valued input out : ndarray, optional Optional output array for function values Returns ------- scalar or ndarray Value of the inverse of the regularized incomplete beta function See Also -------- betainc : regularized incomplete beta function gamma : gamma function References ---------- .. [1] NIST Digital Library of Mathematical Functions https://dlmf.nist.gov/8.17 Examples -------- >>> import scipy.special as sc This function is the inverse of `betainc` for fixed values of :math:`a` and :math:`b`. >>> a, b = 1.2, 3.1 >>> y = sc.betainc(a, b, 0.2) >>> sc.betaincinv(a, b, y) 0.2 >>> >>> a, b = 7.5, 0.4 >>> x = sc.betaincinv(a, b, 0.5) >>> sc.betainc(a, b, x) 0.5betalnbetaln(a, b, out=None) Natural logarithm of absolute value of beta function. Computes ``ln(abs(beta(a, b)))``. Parameters ---------- a, b : array_like Positive, real-valued parameters out : ndarray, optional Optional output array for function values Returns ------- scalar or ndarray Value of the betaln function See Also -------- gamma : the gamma function betainc : the regularized incomplete beta function beta : the beta function Examples -------- >>> import numpy as np >>> from scipy.special import betaln, beta Verify that, for moderate values of ``a`` and ``b``, ``betaln(a, b)`` is the same as ``log(beta(a, b))``: >>> betaln(3, 4) -4.0943445622221 >>> np.log(beta(3, 4)) -4.0943445622221 In the following ``beta(a, b)`` underflows to 0, so we can't compute the logarithm of the actual value. >>> a = 400 >>> b = 900 >>> beta(a, b) 0.0 We can compute the logarithm of ``beta(a, b)`` by using `betaln`: >>> betaln(a, b) -804.3069951764146binombinom(x, y, out=None) Binomial coefficient considered as a function of two real variables. For real arguments, the binomial coefficient is defined as .. math:: \binom{x}{y} = \frac{\Gamma(x + 1)}{\Gamma(y + 1)\Gamma(x - y + 1)} = \frac{1}{(x + 1)\mathrm{B}(x - y + 1, y + 1)} Where :math:`\Gamma` is the Gamma function (`gamma`) and :math:`\mathrm{B}` is the Beta function (`beta`) [1]_. Parameters ---------- x, y: array_like Real arguments to :math:`\binom{x}{y}`. out : ndarray, optional Optional output array for the function values Returns ------- scalar or ndarray Value of binomial coefficient. See Also -------- comb : The number of combinations of N things taken k at a time. Notes ----- The Gamma function has poles at non-positive integers and tends to either positive or negative infinity depending on the direction on the real line from which a pole is approached. When considered as a function of two real variables, :math:`\binom{x}{y}` is thus undefined when `x` is a negative integer. `binom` returns ``nan`` when ``x`` is a negative integer. This is the case even when ``x`` is a negative integer and ``y`` an integer, contrary to the usual convention for defining :math:`\binom{n}{k}` when it is considered as a function of two integer variables. References ---------- .. [1] https://en.wikipedia.org/wiki/Binomial_coefficient Examples -------- The following examples illustrate the ways in which `binom` differs from the function `comb`. >>> from scipy.special import binom, comb When ``exact=False`` and ``x`` and ``y`` are both positive, `comb` calls `binom` internally. >>> x, y = 3, 2 >>> (binom(x, y), comb(x, y), comb(x, y, exact=True)) (3.0, 3.0, 3) For larger values, `comb` with ``exact=True`` no longer agrees with `binom`. >>> x, y = 43, 23 >>> (binom(x, y), comb(x, y), comb(x, y, exact=True)) (960566918219.9999, 960566918219.9999, 960566918220) `binom` returns ``nan`` when ``x`` is a negative integer, but is otherwise defined for negative arguments. `comb` returns 0 whenever one of ``x`` or ``y`` is negative or ``x`` is less than ``y``. >>> x, y = -3, 2 >>> (binom(x, y), comb(x, y), comb(x, y, exact=True)) (nan, 0.0, 0) >>> x, y = -3.1, 2.2 >>> (binom(x, y), comb(x, y), comb(x, y, exact=True)) (18.714147876804432, 0.0, 0) >>> x, y = 2.2, 3.1 >>> (binom(x, y), comb(x, y), comb(x, y, exact=True)) (0.037399983365134115, 0.0, 0)boxcoxboxcox(x, lmbda, out=None) Compute the Box-Cox transformation. The Box-Cox transformation is:: y = (x**lmbda - 1) / lmbda if lmbda != 0 log(x) if lmbda == 0 Returns `nan` if ``x < 0``. Returns `-inf` if ``x == 0`` and ``lmbda < 0``. Parameters ---------- x : array_like Data to be transformed. lmbda : array_like Power parameter of the Box-Cox transform. out : ndarray, optional Optional output array for the function values Returns ------- y : scalar or ndarray Transformed data. Notes ----- .. versionadded:: 0.14.0 Examples -------- >>> from scipy.special import boxcox >>> boxcox([1, 4, 10], 2.5) array([ 0. , 12.4 , 126.09110641]) >>> boxcox(2, [0, 1, 2]) array([ 0.69314718, 1. , 1.5 ])boxcox1pboxcox1p(x, lmbda, out=None) Compute the Box-Cox transformation of 1 + `x`. The Box-Cox transformation computed by `boxcox1p` is:: y = ((1+x)**lmbda - 1) / lmbda if lmbda != 0 log(1+x) if lmbda == 0 Returns `nan` if ``x < -1``. Returns `-inf` if ``x == -1`` and ``lmbda < 0``. Parameters ---------- x : array_like Data to be transformed. lmbda : array_like Power parameter of the Box-Cox transform. out : ndarray, optional Optional output array for the function values Returns ------- y : scalar or ndarray Transformed data. Notes ----- .. versionadded:: 0.14.0 Examples -------- >>> from scipy.special import boxcox1p >>> boxcox1p(1e-4, [0, 0.5, 1]) array([ 9.99950003e-05, 9.99975001e-05, 1.00000000e-04]) >>> boxcox1p([0.01, 0.1], 0.25) array([ 0.00996272, 0.09645476])btdtrbtdtr(a, b, x, out=None) Cumulative distribution function of the beta distribution. Returns the integral from zero to `x` of the beta probability density function, .. math:: I = \int_0^x \frac{\Gamma(a + b)}{\Gamma(a)\Gamma(b)} t^{a-1} (1-t)^{b-1}\,dt where :math:`\Gamma` is the gamma function. Parameters ---------- a : array_like Shape parameter (a > 0). b : array_like Shape parameter (b > 0). x : array_like Upper limit of integration, in [0, 1]. out : ndarray, optional Optional output array for the function values Returns ------- I : scalar or ndarray Cumulative distribution function of the beta distribution with parameters `a` and `b` at `x`. See Also -------- betainc Notes ----- This function is identical to the incomplete beta integral function `betainc`. Wrapper for the Cephes [1]_ routine `btdtr`. References ---------- .. [1] Cephes Mathematical Functions Library, http://www.netlib.org/cephes/btdtribtdtri(a, b, p, out=None) The `p`-th quantile of the beta distribution. This function is the inverse of the beta cumulative distribution function, `btdtr`, returning the value of `x` for which `btdtr(a, b, x) = p`, or .. math:: p = \int_0^x \frac{\Gamma(a + b)}{\Gamma(a)\Gamma(b)} t^{a-1} (1-t)^{b-1}\,dt Parameters ---------- a : array_like Shape parameter (`a` > 0). b : array_like Shape parameter (`b` > 0). p : array_like Cumulative probability, in [0, 1]. out : ndarray, optional Optional output array for the function values Returns ------- x : scalar or ndarray The quantile corresponding to `p`. See Also -------- betaincinv btdtr Notes ----- The value of `x` is found by interval halving or Newton iterations. Wrapper for the Cephes [1]_ routine `incbi`, which solves the equivalent problem of finding the inverse of the incomplete beta integral. References ---------- .. [1] Cephes Mathematical Functions Library, http://www.netlib.org/cephes/btdtriabtdtria(p, b, x, out=None) Inverse of `btdtr` with respect to `a`. This is the inverse of the beta cumulative distribution function, `btdtr`, considered as a function of `a`, returning the value of `a` for which `btdtr(a, b, x) = p`, or .. math:: p = \int_0^x \frac{\Gamma(a + b)}{\Gamma(a)\Gamma(b)} t^{a-1} (1-t)^{b-1}\,dt Parameters ---------- p : array_like Cumulative probability, in [0, 1]. b : array_like Shape parameter (`b` > 0). x : array_like The quantile, in [0, 1]. out : ndarray, optional Optional output array for the function values Returns ------- a : scalar or ndarray The value of the shape parameter `a` such that `btdtr(a, b, x) = p`. See Also -------- btdtr : Cumulative distribution function of the beta distribution. btdtri : Inverse with respect to `x`. btdtrib : Inverse with respect to `b`. Notes ----- Wrapper for the CDFLIB [1]_ Fortran routine `cdfbet`. The cumulative distribution function `p` is computed using a routine by DiDinato and Morris [2]_. Computation of `a` involves a search for a value that produces the desired value of `p`. The search relies on the monotonicity of `p` with `a`. References ---------- .. [1] Barry Brown, James Lovato, and Kathy Russell, CDFLIB: Library of Fortran Routines for Cumulative Distribution Functions, Inverses, and Other Parameters. .. [2] DiDinato, A. R. and Morris, A. H., Algorithm 708: Significant Digit Computation of the Incomplete Beta Function Ratios. ACM Trans. Math. Softw. 18 (1993), 360-373.btdtribbtdtria(a, p, x, out=None) Inverse of `btdtr` with respect to `b`. This is the inverse of the beta cumulative distribution function, `btdtr`, considered as a function of `b`, returning the value of `b` for which `btdtr(a, b, x) = p`, or .. math:: p = \int_0^x \frac{\Gamma(a + b)}{\Gamma(a)\Gamma(b)} t^{a-1} (1-t)^{b-1}\,dt Parameters ---------- a : array_like Shape parameter (`a` > 0). p : array_like Cumulative probability, in [0, 1]. x : array_like The quantile, in [0, 1]. out : ndarray, optional Optional output array for the function values Returns ------- b : scalar or ndarray The value of the shape parameter `b` such that `btdtr(a, b, x) = p`. See Also -------- btdtr : Cumulative distribution function of the beta distribution. btdtri : Inverse with respect to `x`. btdtria : Inverse with respect to `a`. Notes ----- Wrapper for the CDFLIB [1]_ Fortran routine `cdfbet`. The cumulative distribution function `p` is computed using a routine by DiDinato and Morris [2]_. Computation of `b` involves a search for a value that produces the desired value of `p`. The search relies on the monotonicity of `p` with `b`. References ---------- .. [1] Barry Brown, James Lovato, and Kathy Russell, CDFLIB: Library of Fortran Routines for Cumulative Distribution Functions, Inverses, and Other Parameters. .. [2] DiDinato, A. R. and Morris, A. H., Algorithm 708: Significant Digit Computation of the Incomplete Beta Function Ratios. ACM Trans. Math. Softw. 18 (1993), 360-373.cbrtcbrt(x, out=None) Element-wise cube root of `x`. Parameters ---------- x : array_like `x` must contain real numbers. out : ndarray, optional Optional output array for the function values Returns ------- scalar or ndarray The cube root of each value in `x`. Examples -------- >>> from scipy.special import cbrt >>> cbrt(8) 2.0 >>> cbrt([-8, -3, 0.125, 1.331]) array([-2. , -1.44224957, 0.5 , 1.1 ])chdtrchdtr(v, x, out=None) Chi square cumulative distribution function. Returns the area under the left tail (from 0 to `x`) of the Chi square probability density function with `v` degrees of freedom: .. math:: \frac{1}{2^{v/2} \Gamma(v/2)} \int_0^x t^{v/2 - 1} e^{-t/2} dt Here :math:`\Gamma` is the Gamma function; see `gamma`. This integral can be expressed in terms of the regularized lower incomplete gamma function `gammainc` as ``gammainc(v / 2, x / 2)``. [1]_ Parameters ---------- v : array_like Degrees of freedom. x : array_like Upper bound of the integral. out : ndarray, optional Optional output array for the function results. Returns ------- scalar or ndarray Values of the cumulative distribution function. See Also -------- chdtrc, chdtri, chdtriv, gammainc References ---------- .. [1] Chi-Square distribution, https://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm Examples -------- >>> import numpy as np >>> import scipy.special as sc It can be expressed in terms of the regularized lower incomplete gamma function. >>> v = 1 >>> x = np.arange(4) >>> sc.chdtr(v, x) array([0. , 0.68268949, 0.84270079, 0.91673548]) >>> sc.gammainc(v / 2, x / 2) array([0. , 0.68268949, 0.84270079, 0.91673548])chdtrcchdtrc(v, x, out=None) Chi square survival function. Returns the area under the right hand tail (from `x` to infinity) of the Chi square probability density function with `v` degrees of freedom: .. math:: \frac{1}{2^{v/2} \Gamma(v/2)} \int_x^\infty t^{v/2 - 1} e^{-t/2} dt Here :math:`\Gamma` is the Gamma function; see `gamma`. This integral can be expressed in terms of the regularized upper incomplete gamma function `gammaincc` as ``gammaincc(v / 2, x / 2)``. [1]_ Parameters ---------- v : array_like Degrees of freedom. x : array_like Lower bound of the integral. out : ndarray, optional Optional output array for the function results. Returns ------- scalar or ndarray Values of the survival function. See Also -------- chdtr, chdtri, chdtriv, gammaincc References ---------- .. [1] Chi-Square distribution, https://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm Examples -------- >>> import numpy as np >>> import scipy.special as sc It can be expressed in terms of the regularized upper incomplete gamma function. >>> v = 1 >>> x = np.arange(4) >>> sc.chdtrc(v, x) array([1. , 0.31731051, 0.15729921, 0.08326452]) >>> sc.gammaincc(v / 2, x / 2) array([1. , 0.31731051, 0.15729921, 0.08326452])chdtrichdtri(v, p, out=None) Inverse to `chdtrc` with respect to `x`. Returns `x` such that ``chdtrc(v, x) == p``. Parameters ---------- v : array_like Degrees of freedom. p : array_like Probability. out : ndarray, optional Optional output array for the function results. Returns ------- x : scalar or ndarray Value so that the probability a Chi square random variable with `v` degrees of freedom is greater than `x` equals `p`. See Also -------- chdtrc, chdtr, chdtriv References ---------- .. [1] Chi-Square distribution, https://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm Examples -------- >>> import scipy.special as sc It inverts `chdtrc`. >>> v, p = 1, 0.3 >>> sc.chdtrc(v, sc.chdtri(v, p)) 0.3 >>> x = 1 >>> sc.chdtri(v, sc.chdtrc(v, x)) 1.0chdtrivchdtriv(p, x, out=None) Inverse to `chdtr` with respect to `v`. Returns `v` such that ``chdtr(v, x) == p``. Parameters ---------- p : array_like Probability that the Chi square random variable is less than or equal to `x`. x : array_like Nonnegative input. out : ndarray, optional Optional output array for the function results. Returns ------- scalar or ndarray Degrees of freedom. See Also -------- chdtr, chdtrc, chdtri References ---------- .. [1] Chi-Square distribution, https://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm Examples -------- >>> import scipy.special as sc It inverts `chdtr`. >>> p, x = 0.5, 1 >>> sc.chdtr(sc.chdtriv(p, x), x) 0.5000000000202172 >>> v = 1 >>> sc.chdtriv(sc.chdtr(v, x), v) 1.0000000000000013chndtrchndtr(x, df, nc, out=None) Non-central chi square cumulative distribution function The cumulative distribution function is given by: .. math:: P(\chi^{\prime 2} \vert \nu, \lambda) =\sum_{j=0}^{\infty} e^{-\lambda /2} \frac{(\lambda /2)^j}{j!} P(\chi^{\prime 2} \vert \nu + 2j), where :math:`\nu > 0` is the degrees of freedom (``df``) and :math:`\lambda \geq 0` is the non-centrality parameter (``nc``). Parameters ---------- x : array_like Upper bound of the integral; must satisfy ``x >= 0`` df : array_like Degrees of freedom; must satisfy ``df > 0`` nc : array_like Non-centrality parameter; must satisfy ``nc >= 0`` out : ndarray, optional Optional output array for the function results Returns ------- x : scalar or ndarray Value of the non-central chi square cumulative distribution function. See Also -------- chndtrix, chndtridf, chndtrincchndtridfchndtridf(x, p, nc, out=None) Inverse to `chndtr` vs `df` Calculated using a search to find a value for `df` that produces the desired value of `p`. Parameters ---------- x : array_like Upper bound of the integral; must satisfy ``x >= 0`` p : array_like Probability; must satisfy ``0 <= p < 1`` nc : array_like Non-centrality parameter; must satisfy ``nc >= 0`` out : ndarray, optional Optional output array for the function results Returns ------- df : scalar or ndarray Degrees of freedom See Also -------- chndtr, chndtrix, chndtrincchndtrinc(x, df, p, out=None) Inverse to `chndtr` vs `nc` Calculated using a search to find a value for `df` that produces the desired value of `p`. Parameters ---------- x : array_like Upper bound of the integral; must satisfy ``x >= 0`` df : array_like Degrees of freedom; must satisfy ``df > 0`` p : array_like Probability; must satisfy ``0 <= p < 1`` out : ndarray, optional Optional output array for the function results Returns ------- nc : scalar or ndarray Non-centrality See Also -------- chndtr, chndtrix, chndtrincchndtrixchndtrix(p, df, nc, out=None) Inverse to `chndtr` vs `x` Calculated using a search to find a value for `x` that produces the desired value of `p`. Parameters ---------- p : array_like Probability; must satisfy ``0 <= p < 1`` df : array_like Degrees of freedom; must satisfy ``df > 0`` nc : array_like Non-centrality parameter; must satisfy ``nc >= 0`` out : ndarray, optional Optional output array for the function results Returns ------- x : scalar or ndarray Value so that the probability a non-central Chi square random variable with `df` degrees of freedom and non-centrality, `nc`, is greater than `x` equals `p`. See Also -------- chndtr, chndtridf, chndtrinccosdgcosdg(x, out=None) Cosine of the angle `x` given in degrees. Parameters ---------- x : array_like Angle, given in degrees. out : ndarray, optional Optional output array for the function results. Returns ------- scalar or ndarray Cosine of the input. See Also -------- sindg, tandg, cotdg Examples -------- >>> import numpy as np >>> import scipy.special as sc It is more accurate than using cosine directly. >>> x = 90 + 180 * np.arange(3) >>> sc.cosdg(x) array([-0., 0., -0.]) >>> np.cos(x * np.pi / 180) array([ 6.1232340e-17, -1.8369702e-16, 3.0616170e-16])cosm1cosm1(x, out=None) cos(x) - 1 for use when `x` is near zero. Parameters ---------- x : array_like Real valued argument. out : ndarray, optional Optional output array for the function results. Returns ------- scalar or ndarray Values of ``cos(x) - 1``. See Also -------- expm1, log1p Examples -------- >>> import numpy as np >>> import scipy.special as sc It is more accurate than computing ``cos(x) - 1`` directly for ``x`` around 0. >>> x = 1e-30 >>> np.cos(x) - 1 0.0 >>> sc.cosm1(x) -5.0000000000000005e-61cotdgcotdg(x, out=None) Cotangent of the angle `x` given in degrees. Parameters ---------- x : array_like Angle, given in degrees. out : ndarray, optional Optional output array for the function results. Returns ------- scalar or ndarray Cotangent at the input. See Also -------- sindg, cosdg, tandg Examples -------- >>> import numpy as np >>> import scipy.special as sc It is more accurate than using cotangent directly. >>> x = 90 + 180 * np.arange(3) >>> sc.cotdg(x) array([0., 0., 0.]) >>> 1 / np.tan(x * np.pi / 180) array([6.1232340e-17, 1.8369702e-16, 3.0616170e-16])dawsndawsn(x, out=None) Dawson's integral. Computes:: exp(-x**2) * integral(exp(t**2), t=0..x). Parameters ---------- x : array_like Function parameter. out : ndarray, optional Optional output array for the function values Returns ------- y : scalar or ndarray Value of the integral. See Also -------- wofz, erf, erfc, erfcx, erfi References ---------- .. [1] Steven G. Johnson, Faddeeva W function implementation. http://ab-initio.mit.edu/Faddeeva Examples -------- >>> import numpy as np >>> from scipy import special >>> import matplotlib.pyplot as plt >>> x = np.linspace(-15, 15, num=1000) >>> plt.plot(x, special.dawsn(x)) >>> plt.xlabel('$x$') >>> plt.ylabel('$dawsn(x)$') >>> plt.show()ellipeellipe(m, out=None) Complete elliptic integral of the second kind This function is defined as .. math:: E(m) = \int_0^{\pi/2} [1 - m \sin(t)^2]^{1/2} dt Parameters ---------- m : array_like Defines the parameter of the elliptic integral. out : ndarray, optional Optional output array for the function values Returns ------- E : scalar or ndarray Value of the elliptic integral. Notes ----- Wrapper for the Cephes [1]_ routine `ellpe`. For `m > 0` the computation uses the approximation, .. math:: E(m) \approx P(1-m) - (1-m) \log(1-m) Q(1-m), where :math:`P` and :math:`Q` are tenth-order polynomials. For `m < 0`, the relation .. math:: E(m) = E(m/(m - 1)) \sqrt(1-m) is used. The parameterization in terms of :math:`m` follows that of section 17.2 in [2]_. Other parameterizations in terms of the complementary parameter :math:`1 - m`, modular angle :math:`\sin^2(\alpha) = m`, or modulus :math:`k^2 = m` are also used, so be careful that you choose the correct parameter. The Legendre E integral is related to Carlson's symmetric R_D or R_G functions in multiple ways [3]_. For example, .. math:: E(m) = 2 R_G(0, 1-k^2, 1) . See Also -------- ellipkm1 : Complete elliptic integral of the first kind, near `m` = 1 ellipk : Complete elliptic integral of the first kind ellipkinc : Incomplete elliptic integral of the first kind ellipeinc : Incomplete elliptic integral of the second kind elliprd : Symmetric elliptic integral of the second kind. elliprg : Completely-symmetric elliptic integral of the second kind. References ---------- .. [1] Cephes Mathematical Functions Library, http://www.netlib.org/cephes/ .. [2] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. .. [3] NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release 1.0.28 of 2020-09-15. See Sec. 19.25(i) https://dlmf.nist.gov/19.25#i Examples -------- This function is used in finding the circumference of an ellipse with semi-major axis `a` and semi-minor axis `b`. >>> import numpy as np >>> from scipy import special >>> a = 3.5 >>> b = 2.1 >>> e_sq = 1.0 - b**2/a**2 # eccentricity squared Then the circumference is found using the following: >>> C = 4*a*special.ellipe(e_sq) # circumference formula >>> C 17.868899204378693 When `a` and `b` are the same (meaning eccentricity is 0), this reduces to the circumference of a circle. >>> 4*a*special.ellipe(0.0) # formula for ellipse with a = b 21.991148575128552 >>> 2*np.pi*a # formula for circle of radius a 21.991148575128552ellipeincellipeinc(phi, m, out=None) Incomplete elliptic integral of the second kind This function is defined as .. math:: E(\phi, m) = \int_0^{\phi} [1 - m \sin(t)^2]^{1/2} dt Parameters ---------- phi : array_like amplitude of the elliptic integral. m : array_like parameter of the elliptic integral. out : ndarray, optional Optional output array for the function values Returns ------- E : scalar or ndarray Value of the elliptic integral. Notes ----- Wrapper for the Cephes [1]_ routine `ellie`. Computation uses arithmetic-geometric means algorithm. The parameterization in terms of :math:`m` follows that of section 17.2 in [2]_. Other parameterizations in terms of the complementary parameter :math:`1 - m`, modular angle :math:`\sin^2(\alpha) = m`, or modulus :math:`k^2 = m` are also used, so be careful that you choose the correct parameter. The Legendre E incomplete integral can be related to combinations of Carlson's symmetric integrals R_D, R_F, and R_G in multiple ways [3]_. For example, with :math:`c = \csc^2\phi`, .. math:: E(\phi, m) = R_F(c-1, c-k^2, c) - \frac{1}{3} k^2 R_D(c-1, c-k^2, c) . See Also -------- ellipkm1 : Complete elliptic integral of the first kind, near `m` = 1 ellipk : Complete elliptic integral of the first kind ellipkinc : Incomplete elliptic integral of the first kind ellipe : Complete elliptic integral of the second kind elliprd : Symmetric elliptic integral of the second kind. elliprf : Completely-symmetric elliptic integral of the first kind. elliprg : Completely-symmetric elliptic integral of the second kind. References ---------- .. [1] Cephes Mathematical Functions Library, http://www.netlib.org/cephes/ .. [2] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. .. [3] NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release 1.0.28 of 2020-09-15. See Sec. 19.25(i) https://dlmf.nist.gov/19.25#iellipjellipj(u, m, out=None) Jacobian elliptic functions Calculates the Jacobian elliptic functions of parameter `m` between 0 and 1, and real argument `u`. Parameters ---------- m : array_like Parameter. u : array_like Argument. out : tuple of ndarray, optional Optional output arrays for the function values Returns ------- sn, cn, dn, ph : 4-tuple of scalar or ndarray The returned functions:: sn(u|m), cn(u|m), dn(u|m) The value `ph` is such that if `u = ellipkinc(ph, m)`, then `sn(u|m) = sin(ph)` and `cn(u|m) = cos(ph)`. Notes ----- Wrapper for the Cephes [1]_ routine `ellpj`. These functions are periodic, with quarter-period on the real axis equal to the complete elliptic integral `ellipk(m)`. Relation to incomplete elliptic integral: If `u = ellipkinc(phi,m)`, then `sn(u|m) = sin(phi)`, and `cn(u|m) = cos(phi)`. The `phi` is called the amplitude of `u`. Computation is by means of the arithmetic-geometric mean algorithm, except when `m` is within 1e-9 of 0 or 1. In the latter case with `m` close to 1, the approximation applies only for `phi < pi/2`. See also -------- ellipk : Complete elliptic integral of the first kind ellipkinc : Incomplete elliptic integral of the first kind References ---------- .. [1] Cephes Mathematical Functions Library, http://www.netlib.org/cephes/ellipkellipk(m, out=None) Complete elliptic integral of the first kind. This function is defined as .. math:: K(m) = \int_0^{\pi/2} [1 - m \sin(t)^2]^{-1/2} dt Parameters ---------- m : array_like The parameter of the elliptic integral. out : ndarray, optional Optional output array for the function values Returns ------- K : scalar or ndarray Value of the elliptic integral. Notes ----- For more precision around point m = 1, use `ellipkm1`, which this function calls. The parameterization in terms of :math:`m` follows that of section 17.2 in [1]_. Other parameterizations in terms of the complementary parameter :math:`1 - m`, modular angle :math:`\sin^2(\alpha) = m`, or modulus :math:`k^2 = m` are also used, so be careful that you choose the correct parameter. The Legendre K integral is related to Carlson's symmetric R_F function by [2]_: .. math:: K(m) = R_F(0, 1-k^2, 1) . See Also -------- ellipkm1 : Complete elliptic integral of the first kind around m = 1 ellipkinc : Incomplete elliptic integral of the first kind ellipe : Complete elliptic integral of the second kind ellipeinc : Incomplete elliptic integral of the second kind elliprf : Completely-symmetric elliptic integral of the first kind. References ---------- .. [1] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. .. [2] NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release 1.0.28 of 2020-09-15. See Sec. 19.25(i) https://dlmf.nist.gov/19.25#iellipkincellipkinc(phi, m, out=None) Incomplete elliptic integral of the first kind This function is defined as .. math:: K(\phi, m) = \int_0^{\phi} [1 - m \sin(t)^2]^{-1/2} dt This function is also called :math:`F(\phi, m)`. Parameters ---------- phi : array_like amplitude of the elliptic integral m : array_like parameter of the elliptic integral out : ndarray, optional Optional output array for the function values Returns ------- K : scalar or ndarray Value of the elliptic integral Notes ----- Wrapper for the Cephes [1]_ routine `ellik`. The computation is carried out using the arithmetic-geometric mean algorithm. The parameterization in terms of :math:`m` follows that of section 17.2 in [2]_. Other parameterizations in terms of the complementary parameter :math:`1 - m`, modular angle :math:`\sin^2(\alpha) = m`, or modulus :math:`k^2 = m` are also used, so be careful that you choose the correct parameter. The Legendre K incomplete integral (or F integral) is related to Carlson's symmetric R_F function [3]_. Setting :math:`c = \csc^2\phi`, .. math:: F(\phi, m) = R_F(c-1, c-k^2, c) . See Also -------- ellipkm1 : Complete elliptic integral of the first kind, near `m` = 1 ellipk : Complete elliptic integral of the first kind ellipe : Complete elliptic integral of the second kind ellipeinc : Incomplete elliptic integral of the second kind elliprf : Completely-symmetric elliptic integral of the first kind. References ---------- .. [1] Cephes Mathematical Functions Library, http://www.netlib.org/cephes/ .. [2] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. .. [3] NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release 1.0.28 of 2020-09-15. See Sec. 19.25(i) https://dlmf.nist.gov/19.25#iellipkm1ellipkm1(p, out=None) Complete elliptic integral of the first kind around `m` = 1 This function is defined as .. math:: K(p) = \int_0^{\pi/2} [1 - m \sin(t)^2]^{-1/2} dt where `m = 1 - p`. Parameters ---------- p : array_like Defines the parameter of the elliptic integral as `m = 1 - p`. out : ndarray, optional Optional output array for the function values Returns ------- K : scalar or ndarray Value of the elliptic integral. Notes ----- Wrapper for the Cephes [1]_ routine `ellpk`. For `p <= 1`, computation uses the approximation, .. math:: K(p) \approx P(p) - \log(p) Q(p), where :math:`P` and :math:`Q` are tenth-order polynomials. The argument `p` is used internally rather than `m` so that the logarithmic singularity at `m = 1` will be shifted to the origin; this preserves maximum accuracy. For `p > 1`, the identity .. math:: K(p) = K(1/p)/\sqrt(p) is used. See Also -------- ellipk : Complete elliptic integral of the first kind ellipkinc : Incomplete elliptic integral of the first kind ellipe : Complete elliptic integral of the second kind ellipeinc : Incomplete elliptic integral of the second kind elliprf : Completely-symmetric elliptic integral of the first kind. References ---------- .. [1] Cephes Mathematical Functions Library, http://www.netlib.org/cephes/elliprcelliprc(x, y, out=None) Degenerate symmetric elliptic integral. The function RC is defined as [1]_ .. math:: R_{\mathrm{C}}(x, y) = \frac{1}{2} \int_0^{+\infty} (t + x)^{-1/2} (t + y)^{-1} dt = R_{\mathrm{F}}(x, y, y) Parameters ---------- x, y : array_like Real or complex input parameters. `x` can be any number in the complex plane cut along the negative real axis. `y` must be non-zero. out : ndarray, optional Optional output array for the function values Returns ------- R : scalar or ndarray Value of the integral. If `y` is real and negative, the Cauchy principal value is returned. If both of `x` and `y` are real, the return value is real. Otherwise, the return value is complex. Notes ----- RC is a degenerate case of the symmetric integral RF: ``elliprc(x, y) == elliprf(x, y, y)``. It is an elementary function rather than an elliptic integral. The code implements Carlson's algorithm based on the duplication theorems and series expansion up to the 7th order. [2]_ .. versionadded:: 1.8.0 See Also -------- elliprf : Completely-symmetric elliptic integral of the first kind. elliprd : Symmetric elliptic integral of the second kind. elliprg : Completely-symmetric elliptic integral of the second kind. elliprj : Symmetric elliptic integral of the third kind. References ---------- .. [1] B. C. Carlson, ed., Chapter 19 in "Digital Library of Mathematical Functions," NIST, US Dept. of Commerce. https://dlmf.nist.gov/19.16.E6 .. [2] B. C. Carlson, "Numerical computation of real or complex elliptic integrals," Numer. Algorithm, vol. 10, no. 1, pp. 13-26, 1995. https://arxiv.org/abs/math/9409227 https://doi.org/10.1007/BF02198293 Examples -------- Basic homogeneity property: >>> import numpy as np >>> from scipy.special import elliprc >>> x = 1.2 + 3.4j >>> y = 5. >>> scale = 0.3 + 0.4j >>> elliprc(scale*x, scale*y) (0.5484493976710874-0.4169557678995833j) >>> elliprc(x, y)/np.sqrt(scale) (0.5484493976710874-0.41695576789958333j) When the two arguments coincide, the integral is particularly simple: >>> x = 1.2 + 3.4j >>> elliprc(x, x) (0.4299173120614631-0.3041729818745595j) >>> 1/np.sqrt(x) (0.4299173120614631-0.30417298187455954j) Another simple case: the first argument vanishes: >>> y = 1.2 + 3.4j >>> elliprc(0, y) (0.6753125346116815-0.47779380263880866j) >>> np.pi/2/np.sqrt(y) (0.6753125346116815-0.4777938026388088j) When `x` and `y` are both positive, we can express :math:`R_C(x,y)` in terms of more elementary functions. For the case :math:`0 \le x < y`, >>> x = 3.2 >>> y = 6. >>> elliprc(x, y) 0.44942991498453444 >>> np.arctan(np.sqrt((y-x)/x))/np.sqrt(y-x) 0.44942991498453433 And for the case :math:`0 \le y < x`, >>> x = 6. >>> y = 3.2 >>> elliprc(x,y) 0.4989837501576147 >>> np.log((np.sqrt(x)+np.sqrt(x-y))/np.sqrt(y))/np.sqrt(x-y) 0.49898375015761476elliprdelliprd(x, y, z, out=None) Symmetric elliptic integral of the second kind. The function RD is defined as [1]_ .. math:: R_{\mathrm{D}}(x, y, z) = \frac{3}{2} \int_0^{+\infty} [(t + x) (t + y)]^{-1/2} (t + z)^{-3/2} dt Parameters ---------- x, y, z : array_like Real or complex input parameters. `x` or `y` can be any number in the complex plane cut along the negative real axis, but at most one of them can be zero, while `z` must be non-zero. out : ndarray, optional Optional output array for the function values Returns ------- R : scalar or ndarray Value of the integral. If all of `x`, `y`, and `z` are real, the return value is real. Otherwise, the return value is complex. Notes ----- RD is a degenerate case of the elliptic integral RJ: ``elliprd(x, y, z) == elliprj(x, y, z, z)``. The code implements Carlson's algorithm based on the duplication theorems and series expansion up to the 7th order. [2]_ .. versionadded:: 1.8.0 See Also -------- elliprc : Degenerate symmetric elliptic integral. elliprf : Completely-symmetric elliptic integral of the first kind. elliprg : Completely-symmetric elliptic integral of the second kind. elliprj : Symmetric elliptic integral of the third kind. References ---------- .. [1] B. C. Carlson, ed., Chapter 19 in "Digital Library of Mathematical Functions," NIST, US Dept. of Commerce. https://dlmf.nist.gov/19.16.E5 .. [2] B. C. Carlson, "Numerical computation of real or complex elliptic integrals," Numer. Algorithm, vol. 10, no. 1, pp. 13-26, 1995. https://arxiv.org/abs/math/9409227 https://doi.org/10.1007/BF02198293 Examples -------- Basic homogeneity property: >>> import numpy as np >>> from scipy.special import elliprd >>> x = 1.2 + 3.4j >>> y = 5. >>> z = 6. >>> scale = 0.3 + 0.4j >>> elliprd(scale*x, scale*y, scale*z) (-0.03703043835680379-0.24500934665683802j) >>> elliprd(x, y, z)*np.power(scale, -1.5) (-0.0370304383568038-0.24500934665683805j) All three arguments coincide: >>> x = 1.2 + 3.4j >>> elliprd(x, x, x) (-0.03986825876151896-0.14051741840449586j) >>> np.power(x, -1.5) (-0.03986825876151894-0.14051741840449583j) The so-called "second lemniscate constant": >>> elliprd(0, 2, 1)/3 0.5990701173677961 >>> from scipy.special import gamma >>> gamma(0.75)**2/np.sqrt(2*np.pi) 0.5990701173677959elliprfelliprf(x, y, z, out=None) Completely-symmetric elliptic integral of the first kind. The function RF is defined as [1]_ .. math:: R_{\mathrm{F}}(x, y, z) = \frac{1}{2} \int_0^{+\infty} [(t + x) (t + y) (t + z)]^{-1/2} dt Parameters ---------- x, y, z : array_like Real or complex input parameters. `x`, `y`, or `z` can be any number in the complex plane cut along the negative real axis, but at most one of them can be zero. out : ndarray, optional Optional output array for the function values Returns ------- R : scalar or ndarray Value of the integral. If all of `x`, `y`, and `z` are real, the return value is real. Otherwise, the return value is complex. Notes ----- The code implements Carlson's algorithm based on the duplication theorems and series expansion up to the 7th order (cf.: https://dlmf.nist.gov/19.36.i) and the AGM algorithm for the complete integral. [2]_ .. versionadded:: 1.8.0 See Also -------- elliprc : Degenerate symmetric integral. elliprd : Symmetric elliptic integral of the second kind. elliprg : Completely-symmetric elliptic integral of the second kind. elliprj : Symmetric elliptic integral of the third kind. References ---------- .. [1] B. C. Carlson, ed., Chapter 19 in "Digital Library of Mathematical Functions," NIST, US Dept. of Commerce. https://dlmf.nist.gov/19.16.E1 .. [2] B. C. Carlson, "Numerical computation of real or complex elliptic integrals," Numer. Algorithm, vol. 10, no. 1, pp. 13-26, 1995. https://arxiv.org/abs/math/9409227 https://doi.org/10.1007/BF02198293 Examples -------- Basic homogeneity property: >>> import numpy as np >>> from scipy.special import elliprf >>> x = 1.2 + 3.4j >>> y = 5. >>> z = 6. >>> scale = 0.3 + 0.4j >>> elliprf(scale*x, scale*y, scale*z) (0.5328051227278146-0.4008623567957094j) >>> elliprf(x, y, z)/np.sqrt(scale) (0.5328051227278147-0.4008623567957095j) All three arguments coincide: >>> x = 1.2 + 3.4j >>> elliprf(x, x, x) (0.42991731206146316-0.30417298187455954j) >>> 1/np.sqrt(x) (0.4299173120614631-0.30417298187455954j) The so-called "first lemniscate constant": >>> elliprf(0, 1, 2) 1.3110287771460598 >>> from scipy.special import gamma >>> gamma(0.25)**2/(4*np.sqrt(2*np.pi)) 1.3110287771460598elliprgelliprg(x, y, z, out=None) Completely-symmetric elliptic integral of the second kind. The function RG is defined as [1]_ .. math:: R_{\mathrm{G}}(x, y, z) = \frac{1}{4} \int_0^{+\infty} [(t + x) (t + y) (t + z)]^{-1/2} \left(\frac{x}{t + x} + \frac{y}{t + y} + \frac{z}{t + z}\right) t dt Parameters ---------- x, y, z : array_like Real or complex input parameters. `x`, `y`, or `z` can be any number in the complex plane cut along the negative real axis. out : ndarray, optional Optional output array for the function values Returns ------- R : scalar or ndarray Value of the integral. If all of `x`, `y`, and `z` are real, the return value is real. Otherwise, the return value is complex. Notes ----- The implementation uses the relation [1]_ .. math:: 2 R_{\mathrm{G}}(x, y, z) = z R_{\mathrm{F}}(x, y, z) - \frac{1}{3} (x - z) (y - z) R_{\mathrm{D}}(x, y, z) + \sqrt{\frac{x y}{z}} and the symmetry of `x`, `y`, `z` when at least one non-zero parameter can be chosen as the pivot. When one of the arguments is close to zero, the AGM method is applied instead. Other special cases are computed following Ref. [2]_ .. versionadded:: 1.8.0 See Also -------- elliprc : Degenerate symmetric integral. elliprd : Symmetric elliptic integral of the second kind. elliprf : Completely-symmetric elliptic integral of the first kind. elliprj : Symmetric elliptic integral of the third kind. References ---------- .. [1] B. C. Carlson, "Numerical computation of real or complex elliptic integrals," Numer. Algorithm, vol. 10, no. 1, pp. 13-26, 1995. https://arxiv.org/abs/math/9409227 https://doi.org/10.1007/BF02198293 .. [2] B. C. Carlson, ed., Chapter 19 in "Digital Library of Mathematical Functions," NIST, US Dept. of Commerce. https://dlmf.nist.gov/19.16.E1 https://dlmf.nist.gov/19.20.ii Examples -------- Basic homogeneity property: >>> import numpy as np >>> from scipy.special import elliprg >>> x = 1.2 + 3.4j >>> y = 5. >>> z = 6. >>> scale = 0.3 + 0.4j >>> elliprg(scale*x, scale*y, scale*z) (1.195936862005246+0.8470988320464167j) >>> elliprg(x, y, z)*np.sqrt(scale) (1.195936862005246+0.8470988320464165j) Simplifications: >>> elliprg(0, y, y) 1.756203682760182 >>> 0.25*np.pi*np.sqrt(y) 1.7562036827601817 >>> elliprg(0, 0, z) 1.224744871391589 >>> 0.5*np.sqrt(z) 1.224744871391589 The surface area of a triaxial ellipsoid with semiaxes ``a``, ``b``, and ``c`` is given by .. math:: S = 4 \pi a b c R_{\mathrm{G}}(1 / a^2, 1 / b^2, 1 / c^2). >>> def ellipsoid_area(a, b, c): ... r = 4.0 * np.pi * a * b * c ... return r * elliprg(1.0 / (a * a), 1.0 / (b * b), 1.0 / (c * c)) >>> print(ellipsoid_area(1, 3, 5)) 108.62688289491807elliprjelliprj(x, y, z, p, out=None) Symmetric elliptic integral of the third kind. The function RJ is defined as [1]_ .. math:: R_{\mathrm{J}}(x, y, z, p) = \frac{3}{2} \int_0^{+\infty} [(t + x) (t + y) (t + z)]^{-1/2} (t + p)^{-1} dt .. warning:: This function should be considered experimental when the inputs are unbalanced. Check correctness with another independent implementation. Parameters ---------- x, y, z, p : array_like Real or complex input parameters. `x`, `y`, or `z` are numbers in the complex plane cut along the negative real axis (subject to further constraints, see Notes), and at most one of them can be zero. `p` must be non-zero. out : ndarray, optional Optional output array for the function values Returns ------- R : scalar or ndarray Value of the integral. If all of `x`, `y`, `z`, and `p` are real, the return value is real. Otherwise, the return value is complex. If `p` is real and negative, while `x`, `y`, and `z` are real, non-negative, and at most one of them is zero, the Cauchy principal value is returned. [1]_ [2]_ Notes ----- The code implements Carlson's algorithm based on the duplication theorems and series expansion up to the 7th order. [3]_ The algorithm is slightly different from its earlier incarnation as it appears in [1]_, in that the call to `elliprc` (or ``atan``/``atanh``, see [4]_) is no longer needed in the inner loop. Asymptotic approximations are used where arguments differ widely in the order of magnitude. [5]_ The input values are subject to certain sufficient but not necessary constaints when input arguments are complex. Notably, ``x``, ``y``, and ``z`` must have non-negative real parts, unless two of them are non-negative and complex-conjugates to each other while the other is a real non-negative number. [1]_ If the inputs do not satisfy the sufficient condition described in Ref. [1]_ they are rejected outright with the output set to NaN. In the case where one of ``x``, ``y``, and ``z`` is equal to ``p``, the function ``elliprd`` should be preferred because of its less restrictive domain. .. versionadded:: 1.8.0 See Also -------- elliprc : Degenerate symmetric integral. elliprd : Symmetric elliptic integral of the second kind. elliprf : Completely-symmetric elliptic integral of the first kind. elliprg : Completely-symmetric elliptic integral of the second kind. References ---------- .. [1] B. C. Carlson, "Numerical computation of real or complex elliptic integrals," Numer. Algorithm, vol. 10, no. 1, pp. 13-26, 1995. https://arxiv.org/abs/math/9409227 https://doi.org/10.1007/BF02198293 .. [2] B. C. Carlson, ed., Chapter 19 in "Digital Library of Mathematical Functions," NIST, US Dept. of Commerce. https://dlmf.nist.gov/19.20.iii .. [3] B. C. Carlson, J. FitzSimmons, "Reduction Theorems for Elliptic Integrands with the Square Root of Two Quadratic Factors," J. Comput. Appl. Math., vol. 118, nos. 1-2, pp. 71-85, 2000. https://doi.org/10.1016/S0377-0427(00)00282-X .. [4] F. Johansson, "Numerical Evaluation of Elliptic Functions, Elliptic Integrals and Modular Forms," in J. Blumlein, C. Schneider, P. Paule, eds., "Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory," pp. 269-293, 2019 (Cham, Switzerland: Springer Nature Switzerland) https://arxiv.org/abs/1806.06725 https://doi.org/10.1007/978-3-030-04480-0 .. [5] B. C. Carlson, J. L. Gustafson, "Asymptotic Approximations for Symmetric Elliptic Integrals," SIAM J. Math. Anls., vol. 25, no. 2, pp. 288-303, 1994. https://arxiv.org/abs/math/9310223 https://doi.org/10.1137/S0036141092228477 Examples -------- Basic homogeneity property: >>> import numpy as np >>> from scipy.special import elliprj >>> x = 1.2 + 3.4j >>> y = 5. >>> z = 6. >>> p = 7. >>> scale = 0.3 - 0.4j >>> elliprj(scale*x, scale*y, scale*z, scale*p) (0.10834905565679157+0.19694950747103812j) >>> elliprj(x, y, z, p)*np.power(scale, -1.5) (0.10834905565679556+0.19694950747103854j) Reduction to simpler elliptic integral: >>> elliprj(x, y, z, z) (0.08288462362195129-0.028376809745123258j) >>> from scipy.special import elliprd >>> elliprd(x, y, z) (0.08288462362195136-0.028376809745123296j) All arguments coincide: >>> elliprj(x, x, x, x) (-0.03986825876151896-0.14051741840449586j) >>> np.power(x, -1.5) (-0.03986825876151894-0.14051741840449583j)entrentr(x, out=None) Elementwise function for computing entropy. .. math:: \text{entr}(x) = \begin{cases} - x \log(x) & x > 0 \\ 0 & x = 0 \\ -\infty & \text{otherwise} \end{cases} Parameters ---------- x : ndarray Input array. out : ndarray, optional Optional output array for the function values Returns ------- res : scalar or ndarray The value of the elementwise entropy function at the given points `x`. See Also -------- kl_div, rel_entr, scipy.stats.entropy Notes ----- .. versionadded:: 0.15.0 This function is concave. The origin of this function is in convex programming; see [1]_. Given a probability distribution :math:`p_1, \ldots, p_n`, the definition of entropy in the context of *information theory* is .. math:: \sum_{i = 1}^n \mathrm{entr}(p_i). To compute the latter quantity, use `scipy.stats.entropy`. References ---------- .. [1] Boyd, Stephen and Lieven Vandenberghe. *Convex optimization*. Cambridge University Press, 2004. :doi:`https://doi.org/10.1017/CBO9780511804441`erferf(z, out=None) Returns the error function of complex argument. It is defined as ``2/sqrt(pi)*integral(exp(-t**2), t=0..z)``. Parameters ---------- x : ndarray Input array. out : ndarray, optional Optional output array for the function values Returns ------- res : scalar or ndarray The values of the error function at the given points `x`. See Also -------- erfc, erfinv, erfcinv, wofz, erfcx, erfi Notes ----- The cumulative of the unit normal distribution is given by ``Phi(z) = 1/2[1 + erf(z/sqrt(2))]``. References ---------- .. [1] https://en.wikipedia.org/wiki/Error_function .. [2] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. http://www.math.sfu.ca/~cbm/aands/page_297.htm .. [3] Steven G. Johnson, Faddeeva W function implementation. http://ab-initio.mit.edu/Faddeeva Examples -------- >>> import numpy as np >>> from scipy import special >>> import matplotlib.pyplot as plt >>> x = np.linspace(-3, 3) >>> plt.plot(x, special.erf(x)) >>> plt.xlabel('$x$') >>> plt.ylabel('$erf(x)$') >>> plt.show()erfcerfc(x, out=None) Complementary error function, ``1 - erf(x)``. Parameters ---------- x : array_like Real or complex valued argument out : ndarray, optional Optional output array for the function results Returns ------- scalar or ndarray Values of the complementary error function See Also -------- erf, erfi, erfcx, dawsn, wofz References ---------- .. [1] Steven G. Johnson, Faddeeva W function implementation. http://ab-initio.mit.edu/Faddeeva Examples -------- >>> import numpy as np >>> from scipy import special >>> import matplotlib.pyplot as plt >>> x = np.linspace(-3, 3) >>> plt.plot(x, special.erfc(x)) >>> plt.xlabel('$x$') >>> plt.ylabel('$erfc(x)$') >>> plt.show()erfcinverfcinv(y, out=None) Inverse of the complementary error function. Computes the inverse of the complementary error function. In the complex domain, there is no unique complex number w satisfying erfc(w)=z. This indicates a true inverse function would be multivalued. When the domain restricts to the real, 0 < x < 2, there is a unique real number satisfying erfc(erfcinv(x)) = erfcinv(erfc(x)). It is related to inverse of the error function by erfcinv(1-x) = erfinv(x) Parameters ---------- y : ndarray Argument at which to evaluate. Domain: [0, 2] out : ndarray, optional Optional output array for the function values Returns ------- erfcinv : scalar or ndarray The inverse of erfc of y, element-wise See Also -------- erf : Error function of a complex argument erfc : Complementary error function, ``1 - erf(x)`` erfinv : Inverse of the error function Examples -------- >>> import numpy as np >>> import matplotlib.pyplot as plt >>> from scipy.special import erfcinv >>> erfcinv(0.5) 0.4769362762044699 >>> y = np.linspace(0.0, 2.0, num=11) >>> erfcinv(y) array([ inf, 0.9061938 , 0.59511608, 0.37080716, 0.17914345, -0. , -0.17914345, -0.37080716, -0.59511608, -0.9061938 , -inf]) Plot the function: >>> y = np.linspace(0, 2, 200) >>> fig, ax = plt.subplots() >>> ax.plot(y, erfcinv(y)) >>> ax.grid(True) >>> ax.set_xlabel('y') >>> ax.set_title('erfcinv(y)') >>> plt.show()erfcxerfcx(x, out=None) Scaled complementary error function, ``exp(x**2) * erfc(x)``. Parameters ---------- x : array_like Real or complex valued argument out : ndarray, optional Optional output array for the function results Returns ------- scalar or ndarray Values of the scaled complementary error function See Also -------- erf, erfc, erfi, dawsn, wofz Notes ----- .. versionadded:: 0.12.0 References ---------- .. [1] Steven G. Johnson, Faddeeva W function implementation. http://ab-initio.mit.edu/Faddeeva Examples -------- >>> import numpy as np >>> from scipy import special >>> import matplotlib.pyplot as plt >>> x = np.linspace(-3, 3) >>> plt.plot(x, special.erfcx(x)) >>> plt.xlabel('$x$') >>> plt.ylabel('$erfcx(x)$') >>> plt.show()erfierfi(z, out=None) Imaginary error function, ``-i erf(i z)``. Parameters ---------- z : array_like Real or complex valued argument out : ndarray, optional Optional output array for the function results Returns ------- scalar or ndarray Values of the imaginary error function See Also -------- erf, erfc, erfcx, dawsn, wofz Notes ----- .. versionadded:: 0.12.0 References ---------- .. [1] Steven G. Johnson, Faddeeva W function implementation. http://ab-initio.mit.edu/Faddeeva Examples -------- >>> import numpy as np >>> from scipy import special >>> import matplotlib.pyplot as plt >>> x = np.linspace(-3, 3) >>> plt.plot(x, special.erfi(x)) >>> plt.xlabel('$x$') >>> plt.ylabel('$erfi(x)$') >>> plt.show()erfinverfinv(y, out=None) Inverse of the error function. Computes the inverse of the error function. In the complex domain, there is no unique complex number w satisfying erf(w)=z. This indicates a true inverse function would be multivalued. When the domain restricts to the real, -1 < x < 1, there is a unique real number satisfying erf(erfinv(x)) = x. Parameters ---------- y : ndarray Argument at which to evaluate. Domain: [-1, 1] out : ndarray, optional Optional output array for the function values Returns ------- erfinv : scalar or ndarray The inverse of erf of y, element-wise See Also -------- erf : Error function of a complex argument erfc : Complementary error function, ``1 - erf(x)`` erfcinv : Inverse of the complementary error function Examples -------- >>> import numpy as np >>> import matplotlib.pyplot as plt >>> from scipy.special import erfinv, erf >>> erfinv(0.5) 0.4769362762044699 >>> y = np.linspace(-1.0, 1.0, num=9) >>> x = erfinv(y) >>> x array([ -inf, -0.81341985, -0.47693628, -0.22531206, 0. , 0.22531206, 0.47693628, 0.81341985, inf]) Verify that ``erf(erfinv(y))`` is ``y``. >>> erf(x) array([-1. , -0.75, -0.5 , -0.25, 0. , 0.25, 0.5 , 0.75, 1. ]) Plot the function: >>> y = np.linspace(-1, 1, 200) >>> fig, ax = plt.subplots() >>> ax.plot(y, erfinv(y)) >>> ax.grid(True) >>> ax.set_xlabel('y') >>> ax.set_title('erfinv(y)') >>> plt.show()eval_chebyceval_chebyc(n, x, out=None) Evaluate Chebyshev polynomial of the first kind on [-2, 2] at a point. These polynomials are defined as .. math:: C_n(x) = 2 T_n(x/2) where :math:`T_n` is a Chebyshev polynomial of the first kind. See 22.5.11 in [AS]_ for details. Parameters ---------- n : array_like Degree of the polynomial. If not an integer, the result is determined via the relation to `eval_chebyt`. x : array_like Points at which to evaluate the Chebyshev polynomial out : ndarray, optional Optional output array for the function values Returns ------- C : scalar or ndarray Values of the Chebyshev polynomial See Also -------- roots_chebyc : roots and quadrature weights of Chebyshev polynomials of the first kind on [-2, 2] chebyc : Chebyshev polynomial object numpy.polynomial.chebyshev.Chebyshev : Chebyshev series eval_chebyt : evaluate Chebycshev polynomials of the first kind References ---------- .. [AS] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. Examples -------- >>> import numpy as np >>> import scipy.special as sc They are a scaled version of the Chebyshev polynomials of the first kind. >>> x = np.linspace(-2, 2, 6) >>> sc.eval_chebyc(3, x) array([-2. , 1.872, 1.136, -1.136, -1.872, 2. ]) >>> 2 * sc.eval_chebyt(3, x / 2) array([-2. , 1.872, 1.136, -1.136, -1.872, 2. ])eval_chebyseval_chebys(n, x, out=None) Evaluate Chebyshev polynomial of the second kind on [-2, 2] at a point. These polynomials are defined as .. math:: S_n(x) = U_n(x/2) where :math:`U_n` is a Chebyshev polynomial of the second kind. See 22.5.13 in [AS]_ for details. Parameters ---------- n : array_like Degree of the polynomial. If not an integer, the result is determined via the relation to `eval_chebyu`. x : array_like Points at which to evaluate the Chebyshev polynomial out : ndarray, optional Optional output array for the function values Returns ------- S : scalar or ndarray Values of the Chebyshev polynomial See Also -------- roots_chebys : roots and quadrature weights of Chebyshev polynomials of the second kind on [-2, 2] chebys : Chebyshev polynomial object eval_chebyu : evaluate Chebyshev polynomials of the second kind References ---------- .. [AS] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. Examples -------- >>> import numpy as np >>> import scipy.special as sc They are a scaled version of the Chebyshev polynomials of the second kind. >>> x = np.linspace(-2, 2, 6) >>> sc.eval_chebys(3, x) array([-4. , 0.672, 0.736, -0.736, -0.672, 4. ]) >>> sc.eval_chebyu(3, x / 2) array([-4. , 0.672, 0.736, -0.736, -0.672, 4. ])eval_chebyteval_chebyt(n, x, out=None) Evaluate Chebyshev polynomial of the first kind at a point. The Chebyshev polynomials of the first kind can be defined via the Gauss hypergeometric function :math:`{}_2F_1` as .. math:: T_n(x) = {}_2F_1(n, -n; 1/2; (1 - x)/2). When :math:`n` is an integer the result is a polynomial of degree :math:`n`. See 22.5.47 in [AS]_ for details. Parameters ---------- n : array_like Degree of the polynomial. If not an integer, the result is determined via the relation to the Gauss hypergeometric function. x : array_like Points at which to evaluate the Chebyshev polynomial out : ndarray, optional Optional output array for the function values Returns ------- T : scalar or ndarray Values of the Chebyshev polynomial See Also -------- roots_chebyt : roots and quadrature weights of Chebyshev polynomials of the first kind chebyu : Chebychev polynomial object eval_chebyu : evaluate Chebyshev polynomials of the second kind hyp2f1 : Gauss hypergeometric function numpy.polynomial.chebyshev.Chebyshev : Chebyshev series Notes ----- This routine is numerically stable for `x` in ``[-1, 1]`` at least up to order ``10000``. References ---------- .. [AS] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972.eval_chebyueval_chebyu(n, x, out=None) Evaluate Chebyshev polynomial of the second kind at a point. The Chebyshev polynomials of the second kind can be defined via the Gauss hypergeometric function :math:`{}_2F_1` as .. math:: U_n(x) = (n + 1) {}_2F_1(-n, n + 2; 3/2; (1 - x)/2). When :math:`n` is an integer the result is a polynomial of degree :math:`n`. See 22.5.48 in [AS]_ for details. Parameters ---------- n : array_like Degree of the polynomial. If not an integer, the result is determined via the relation to the Gauss hypergeometric function. x : array_like Points at which to evaluate the Chebyshev polynomial out : ndarray, optional Optional output array for the function values Returns ------- U : scalar or ndarray Values of the Chebyshev polynomial See Also -------- roots_chebyu : roots and quadrature weights of Chebyshev polynomials of the second kind chebyu : Chebyshev polynomial object eval_chebyt : evaluate Chebyshev polynomials of the first kind hyp2f1 : Gauss hypergeometric function References ---------- .. [AS] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972.eval_gegenbauereval_gegenbauer(n, alpha, x, out=None) Evaluate Gegenbauer polynomial at a point. The Gegenbauer polynomials can be defined via the Gauss hypergeometric function :math:`{}_2F_1` as .. math:: C_n^{(\alpha)} = \frac{(2\alpha)_n}{\Gamma(n + 1)} {}_2F_1(-n, 2\alpha + n; \alpha + 1/2; (1 - z)/2). When :math:`n` is an integer the result is a polynomial of degree :math:`n`. See 22.5.46 in [AS]_ for details. Parameters ---------- n : array_like Degree of the polynomial. If not an integer, the result is determined via the relation to the Gauss hypergeometric function. alpha : array_like Parameter x : array_like Points at which to evaluate the Gegenbauer polynomial out : ndarray, optional Optional output array for the function values Returns ------- C : scalar or ndarray Values of the Gegenbauer polynomial See Also -------- roots_gegenbauer : roots and quadrature weights of Gegenbauer polynomials gegenbauer : Gegenbauer polynomial object hyp2f1 : Gauss hypergeometric function References ---------- .. [AS] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972.eval_genlaguerre(n, alpha, x, out=None) Evaluate generalized Laguerre polynomial at a point. The generalized Laguerre polynomials can be defined via the confluent hypergeometric function :math:`{}_1F_1` as .. math:: L_n^{(\alpha)}(x) = \binom{n + \alpha}{n} {}_1F_1(-n, \alpha + 1, x). When :math:`n` is an integer the result is a polynomial of degree :math:`n`. See 22.5.54 in [AS]_ for details. The Laguerre polynomials are the special case where :math:`\alpha = 0`. Parameters ---------- n : array_like Degree of the polynomial. If not an integer, the result is determined via the relation to the confluent hypergeometric function. alpha : array_like Parameter; must have ``alpha > -1`` x : array_like Points at which to evaluate the generalized Laguerre polynomial out : ndarray, optional Optional output array for the function values Returns ------- L : scalar or ndarray Values of the generalized Laguerre polynomial See Also -------- roots_genlaguerre : roots and quadrature weights of generalized Laguerre polynomials genlaguerre : generalized Laguerre polynomial object hyp1f1 : confluent hypergeometric function eval_laguerre : evaluate Laguerre polynomials References ---------- .. [AS] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972.eval_hermite(n, x, out=None) Evaluate physicist's Hermite polynomial at a point. Defined by .. math:: H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2}; :math:`H_n` is a polynomial of degree :math:`n`. See 22.11.7 in [AS]_ for details. Parameters ---------- n : array_like Degree of the polynomial x : array_like Points at which to evaluate the Hermite polynomial out : ndarray, optional Optional output array for the function values Returns ------- H : scalar or ndarray Values of the Hermite polynomial See Also -------- roots_hermite : roots and quadrature weights of physicist's Hermite polynomials hermite : physicist's Hermite polynomial object numpy.polynomial.hermite.Hermite : Physicist's Hermite series eval_hermitenorm : evaluate Probabilist's Hermite polynomials References ---------- .. [AS] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972.eval_hermitenorm(n, x, out=None) Evaluate probabilist's (normalized) Hermite polynomial at a point. Defined by .. math:: He_n(x) = (-1)^n e^{x^2/2} \frac{d^n}{dx^n} e^{-x^2/2}; :math:`He_n` is a polynomial of degree :math:`n`. See 22.11.8 in [AS]_ for details. Parameters ---------- n : array_like Degree of the polynomial x : array_like Points at which to evaluate the Hermite polynomial out : ndarray, optional Optional output array for the function values Returns ------- He : scalar or ndarray Values of the Hermite polynomial See Also -------- roots_hermitenorm : roots and quadrature weights of probabilist's Hermite polynomials hermitenorm : probabilist's Hermite polynomial object numpy.polynomial.hermite_e.HermiteE : Probabilist's Hermite series eval_hermite : evaluate physicist's Hermite polynomials References ---------- .. [AS] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972.eval_jacobieval_jacobi(n, alpha, beta, x, out=None) Evaluate Jacobi polynomial at a point. The Jacobi polynomials can be defined via the Gauss hypergeometric function :math:`{}_2F_1` as .. math:: P_n^{(\alpha, \beta)}(x) = \frac{(\alpha + 1)_n}{\Gamma(n + 1)} {}_2F_1(-n, 1 + \alpha + \beta + n; \alpha + 1; (1 - z)/2) where :math:`(\cdot)_n` is the Pochhammer symbol; see `poch`. When :math:`n` is an integer the result is a polynomial of degree :math:`n`. See 22.5.42 in [AS]_ for details. Parameters ---------- n : array_like Degree of the polynomial. If not an integer the result is determined via the relation to the Gauss hypergeometric function. alpha : array_like Parameter beta : array_like Parameter x : array_like Points at which to evaluate the polynomial out : ndarray, optional Optional output array for the function values Returns ------- P : scalar or ndarray Values of the Jacobi polynomial See Also -------- roots_jacobi : roots and quadrature weights of Jacobi polynomials jacobi : Jacobi polynomial object hyp2f1 : Gauss hypergeometric function References ---------- .. [AS] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972.eval_laguerreeval_laguerre(n, x, out=None) Evaluate Laguerre polynomial at a point. The Laguerre polynomials can be defined via the confluent hypergeometric function :math:`{}_1F_1` as .. math:: L_n(x) = {}_1F_1(-n, 1, x). See 22.5.16 and 22.5.54 in [AS]_ for details. When :math:`n` is an integer the result is a polynomial of degree :math:`n`. Parameters ---------- n : array_like Degree of the polynomial. If not an integer the result is determined via the relation to the confluent hypergeometric function. x : array_like Points at which to evaluate the Laguerre polynomial out : ndarray, optional Optional output array for the function values Returns ------- L : scalar or ndarray Values of the Laguerre polynomial See Also -------- roots_laguerre : roots and quadrature weights of Laguerre polynomials laguerre : Laguerre polynomial object numpy.polynomial.laguerre.Laguerre : Laguerre series eval_genlaguerre : evaluate generalized Laguerre polynomials References ---------- .. [AS] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972.eval_legendreeval_legendre(n, x, out=None) Evaluate Legendre polynomial at a point. The Legendre polynomials can be defined via the Gauss hypergeometric function :math:`{}_2F_1` as .. math:: P_n(x) = {}_2F_1(-n, n + 1; 1; (1 - x)/2). When :math:`n` is an integer the result is a polynomial of degree :math:`n`. See 22.5.49 in [AS]_ for details. Parameters ---------- n : array_like Degree of the polynomial. If not an integer, the result is determined via the relation to the Gauss hypergeometric function. x : array_like Points at which to evaluate the Legendre polynomial out : ndarray, optional Optional output array for the function values Returns ------- P : scalar or ndarray Values of the Legendre polynomial See Also -------- roots_legendre : roots and quadrature weights of Legendre polynomials legendre : Legendre polynomial object hyp2f1 : Gauss hypergeometric function numpy.polynomial.legendre.Legendre : Legendre series References ---------- .. [AS] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. Examples -------- >>> import numpy as np >>> from scipy.special import eval_legendre Evaluate the zero-order Legendre polynomial at x = 0 >>> eval_legendre(0, 0) 1.0 Evaluate the first-order Legendre polynomial between -1 and 1 >>> X = np.linspace(-1, 1, 5) # Domain of Legendre polynomials >>> eval_legendre(1, X) array([-1. , -0.5, 0. , 0.5, 1. ]) Evaluate Legendre polynomials of order 0 through 4 at x = 0 >>> N = range(0, 5) >>> eval_legendre(N, 0) array([ 1. , 0. , -0.5 , 0. , 0.375]) Plot Legendre polynomials of order 0 through 4 >>> X = np.linspace(-1, 1) >>> import matplotlib.pyplot as plt >>> for n in range(0, 5): ... y = eval_legendre(n, X) ... plt.plot(X, y, label=r'$P_{}(x)$'.format(n)) >>> plt.title("Legendre Polynomials") >>> plt.xlabel("x") >>> plt.ylabel(r'$P_n(x)$') >>> plt.legend(loc='lower right') >>> plt.show()eval_sh_chebyteval_sh_chebyt(n, x, out=None) Evaluate shifted Chebyshev polynomial of the first kind at a point. These polynomials are defined as .. math:: T_n^*(x) = T_n(2x - 1) where :math:`T_n` is a Chebyshev polynomial of the first kind. See 22.5.14 in [AS]_ for details. Parameters ---------- n : array_like Degree of the polynomial. If not an integer, the result is determined via the relation to `eval_chebyt`. x : array_like Points at which to evaluate the shifted Chebyshev polynomial out : ndarray, optional Optional output array for the function values Returns ------- T : scalar or ndarray Values of the shifted Chebyshev polynomial See Also -------- roots_sh_chebyt : roots and quadrature weights of shifted Chebyshev polynomials of the first kind sh_chebyt : shifted Chebyshev polynomial object eval_chebyt : evaluate Chebyshev polynomials of the first kind numpy.polynomial.chebyshev.Chebyshev : Chebyshev series References ---------- .. [AS] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972.eval_sh_chebyueval_sh_chebyu(n, x, out=None) Evaluate shifted Chebyshev polynomial of the second kind at a point. These polynomials are defined as .. math:: U_n^*(x) = U_n(2x - 1) where :math:`U_n` is a Chebyshev polynomial of the first kind. See 22.5.15 in [AS]_ for details. Parameters ---------- n : array_like Degree of the polynomial. If not an integer, the result is determined via the relation to `eval_chebyu`. x : array_like Points at which to evaluate the shifted Chebyshev polynomial out : ndarray, optional Optional output array for the function values Returns ------- U : scalar or ndarray Values of the shifted Chebyshev polynomial See Also -------- roots_sh_chebyu : roots and quadrature weights of shifted Chebychev polynomials of the second kind sh_chebyu : shifted Chebyshev polynomial object eval_chebyu : evaluate Chebyshev polynomials of the second kind References ---------- .. [AS] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972.eval_sh_jacobieval_sh_jacobi(n, p, q, x, out=None) Evaluate shifted Jacobi polynomial at a point. Defined by .. math:: G_n^{(p, q)}(x) = \binom{2n + p - 1}{n}^{-1} P_n^{(p - q, q - 1)}(2x - 1), where :math:`P_n^{(\cdot, \cdot)}` is the n-th Jacobi polynomial. See 22.5.2 in [AS]_ for details. Parameters ---------- n : int Degree of the polynomial. If not an integer, the result is determined via the relation to `binom` and `eval_jacobi`. p : float Parameter q : float Parameter out : ndarray, optional Optional output array for the function values Returns ------- G : scalar or ndarray Values of the shifted Jacobi polynomial. See Also -------- roots_sh_jacobi : roots and quadrature weights of shifted Jacobi polynomials sh_jacobi : shifted Jacobi polynomial object eval_jacobi : evaluate Jacobi polynomials References ---------- .. [AS] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972.eval_sh_legendreeval_sh_legendre(n, x, out=None) Evaluate shifted Legendre polynomial at a point. These polynomials are defined as .. math:: P_n^*(x) = P_n(2x - 1) where :math:`P_n` is a Legendre polynomial. See 2.2.11 in [AS]_ for details. Parameters ---------- n : array_like Degree of the polynomial. If not an integer, the value is determined via the relation to `eval_legendre`. x : array_like Points at which to evaluate the shifted Legendre polynomial out : ndarray, optional Optional output array for the function values Returns ------- P : scalar or ndarray Values of the shifted Legendre polynomial See Also -------- roots_sh_legendre : roots and quadrature weights of shifted Legendre polynomials sh_legendre : shifted Legendre polynomial object eval_legendre : evaluate Legendre polynomials numpy.polynomial.legendre.Legendre : Legendre series References ---------- .. [AS] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972.exp1(z, out=None) Exponential integral E1. For complex :math:`z \ne 0` the exponential integral can be defined as [1]_ .. math:: E_1(z) = \int_z^\infty \frac{e^{-t}}{t} dt, where the path of the integral does not cross the negative real axis or pass through the origin. Parameters ---------- z: array_like Real or complex argument. out : ndarray, optional Optional output array for the function results Returns ------- scalar or ndarray Values of the exponential integral E1 See Also -------- expi : exponential integral :math:`Ei` expn : generalization of :math:`E_1` Notes ----- For :math:`x > 0` it is related to the exponential integral :math:`Ei` (see `expi`) via the relation .. math:: E_1(x) = -Ei(-x). References ---------- .. [1] Digital Library of Mathematical Functions, 6.2.1 https://dlmf.nist.gov/6.2#E1 Examples -------- >>> import numpy as np >>> import scipy.special as sc It has a pole at 0. >>> sc.exp1(0) inf It has a branch cut on the negative real axis. >>> sc.exp1(-1) nan >>> sc.exp1(complex(-1, 0)) (-1.8951178163559368-3.141592653589793j) >>> sc.exp1(complex(-1, -0.0)) (-1.8951178163559368+3.141592653589793j) It approaches 0 along the positive real axis. >>> sc.exp1([1, 10, 100, 1000]) array([2.19383934e-01, 4.15696893e-06, 3.68359776e-46, 0.00000000e+00]) It is related to `expi`. >>> x = np.array([1, 2, 3, 4]) >>> sc.exp1(x) array([0.21938393, 0.04890051, 0.01304838, 0.00377935]) >>> -sc.expi(-x) array([0.21938393, 0.04890051, 0.01304838, 0.00377935])exp10exp10(x, out=None) Compute ``10**x`` element-wise. Parameters ---------- x : array_like `x` must contain real numbers. out : ndarray, optional Optional output array for the function values Returns ------- scalar or ndarray ``10**x``, computed element-wise. Examples -------- >>> import numpy as np >>> from scipy.special import exp10 >>> exp10(3) 1000.0 >>> x = np.array([[-1, -0.5, 0], [0.5, 1, 1.5]]) >>> exp10(x) array([[ 0.1 , 0.31622777, 1. ], [ 3.16227766, 10. , 31.6227766 ]])exp2exp2(x, out=None) Compute ``2**x`` element-wise. Parameters ---------- x : array_like `x` must contain real numbers. out : ndarray, optional Optional output array for the function values Returns ------- scalar or ndarray ``2**x``, computed element-wise. Examples -------- >>> import numpy as np >>> from scipy.special import exp2 >>> exp2(3) 8.0 >>> x = np.array([[-1, -0.5, 0], [0.5, 1, 1.5]]) >>> exp2(x) array([[ 0.5 , 0.70710678, 1. ], [ 1.41421356, 2. , 2.82842712]])expiexpi(x, out=None) Exponential integral Ei. For real :math:`x`, the exponential integral is defined as [1]_ .. math:: Ei(x) = \int_{-\infty}^x \frac{e^t}{t} dt. For :math:`x > 0` the integral is understood as a Cauchy principal value. It is extended to the complex plane by analytic continuation of the function on the interval :math:`(0, \infty)`. The complex variant has a branch cut on the negative real axis. Parameters ---------- x : array_like Real or complex valued argument out : ndarray, optional Optional output array for the function results Returns ------- scalar or ndarray Values of the exponential integral Notes ----- The exponential integrals :math:`E_1` and :math:`Ei` satisfy the relation .. math:: E_1(x) = -Ei(-x) for :math:`x > 0`. See Also -------- exp1 : Exponential integral :math:`E_1` expn : Generalized exponential integral :math:`E_n` References ---------- .. [1] Digital Library of Mathematical Functions, 6.2.5 https://dlmf.nist.gov/6.2#E5 Examples -------- >>> import numpy as np >>> import scipy.special as sc It is related to `exp1`. >>> x = np.array([1, 2, 3, 4]) >>> -sc.expi(-x) array([0.21938393, 0.04890051, 0.01304838, 0.00377935]) >>> sc.exp1(x) array([0.21938393, 0.04890051, 0.01304838, 0.00377935]) The complex variant has a branch cut on the negative real axis. >>> sc.expi(-1 + 1e-12j) (-0.21938393439552062+3.1415926535894254j) >>> sc.expi(-1 - 1e-12j) (-0.21938393439552062-3.1415926535894254j) As the complex variant approaches the branch cut, the real parts approach the value of the real variant. >>> sc.expi(-1) -0.21938393439552062 The SciPy implementation returns the real variant for complex values on the branch cut. >>> sc.expi(complex(-1, 0.0)) (-0.21938393439552062-0j) >>> sc.expi(complex(-1, -0.0)) (-0.21938393439552062-0j)expit(x, out=None) Expit (a.k.a. logistic sigmoid) ufunc for ndarrays. The expit function, also known as the logistic sigmoid function, is defined as ``expit(x) = 1/(1+exp(-x))``. It is the inverse of the logit function. Parameters ---------- x : ndarray The ndarray to apply expit to element-wise. out : ndarray, optional Optional output array for the function values Returns ------- scalar or ndarray An ndarray of the same shape as x. Its entries are `expit` of the corresponding entry of x. See Also -------- logit Notes ----- As a ufunc expit takes a number of optional keyword arguments. For more information see `ufuncs `_ .. versionadded:: 0.10.0 Examples -------- >>> import numpy as np >>> from scipy.special import expit, logit >>> expit([-np.inf, -1.5, 0, 1.5, np.inf]) array([ 0. , 0.18242552, 0.5 , 0.81757448, 1. ]) `logit` is the inverse of `expit`: >>> logit(expit([-2.5, 0, 3.1, 5.0])) array([-2.5, 0. , 3.1, 5. ]) Plot expit(x) for x in [-6, 6]: >>> import matplotlib.pyplot as plt >>> x = np.linspace(-6, 6, 121) >>> y = expit(x) >>> plt.plot(x, y) >>> plt.grid() >>> plt.xlim(-6, 6) >>> plt.xlabel('x') >>> plt.title('expit(x)') >>> plt.show()expm1expm1(x, out=None) Compute ``exp(x) - 1``. When `x` is near zero, ``exp(x)`` is near 1, so the numerical calculation of ``exp(x) - 1`` can suffer from catastrophic loss of precision. ``expm1(x)`` is implemented to avoid the loss of precision that occurs when `x` is near zero. Parameters ---------- x : array_like `x` must contain real numbers. out : ndarray, optional Optional output array for the function values Returns ------- scalar or ndarray ``exp(x) - 1`` computed element-wise. Examples -------- >>> import numpy as np >>> from scipy.special import expm1 >>> expm1(1.0) 1.7182818284590451 >>> expm1([-0.2, -0.1, 0, 0.1, 0.2]) array([-0.18126925, -0.09516258, 0. , 0.10517092, 0.22140276]) The exact value of ``exp(7.5e-13) - 1`` is:: 7.5000000000028125000000007031250000001318...*10**-13. Here is what ``expm1(7.5e-13)`` gives: >>> expm1(7.5e-13) 7.5000000000028135e-13 Compare that to ``exp(7.5e-13) - 1``, where the subtraction results in a "catastrophic" loss of precision: >>> np.exp(7.5e-13) - 1 7.5006667543675576e-13expnexpn(n, x, out=None) Generalized exponential integral En. For integer :math:`n \geq 0` and real :math:`x \geq 0` the generalized exponential integral is defined as [dlmf]_ .. math:: E_n(x) = x^{n - 1} \int_x^\infty \frac{e^{-t}}{t^n} dt. Parameters ---------- n : array_like Non-negative integers x : array_like Real argument out : ndarray, optional Optional output array for the function results Returns ------- scalar or ndarray Values of the generalized exponential integral See Also -------- exp1 : special case of :math:`E_n` for :math:`n = 1` expi : related to :math:`E_n` when :math:`n = 1` References ---------- .. [dlmf] Digital Library of Mathematical Functions, 8.19.2 https://dlmf.nist.gov/8.19#E2 Examples -------- >>> import numpy as np >>> import scipy.special as sc Its domain is nonnegative n and x. >>> sc.expn(-1, 1.0), sc.expn(1, -1.0) (nan, nan) It has a pole at ``x = 0`` for ``n = 1, 2``; for larger ``n`` it is equal to ``1 / (n - 1)``. >>> sc.expn([0, 1, 2, 3, 4], 0) array([ inf, inf, 1. , 0.5 , 0.33333333]) For n equal to 0 it reduces to ``exp(-x) / x``. >>> x = np.array([1, 2, 3, 4]) >>> sc.expn(0, x) array([0.36787944, 0.06766764, 0.01659569, 0.00457891]) >>> np.exp(-x) / x array([0.36787944, 0.06766764, 0.01659569, 0.00457891]) For n equal to 1 it reduces to `exp1`. >>> sc.expn(1, x) array([0.21938393, 0.04890051, 0.01304838, 0.00377935]) >>> sc.exp1(x) array([0.21938393, 0.04890051, 0.01304838, 0.00377935])exprelexprel(x, out=None) Relative error exponential, ``(exp(x) - 1)/x``. When `x` is near zero, ``exp(x)`` is near 1, so the numerical calculation of ``exp(x) - 1`` can suffer from catastrophic loss of precision. ``exprel(x)`` is implemented to avoid the loss of precision that occurs when `x` is near zero. Parameters ---------- x : ndarray Input array. `x` must contain real numbers. out : ndarray, optional Optional output array for the function values Returns ------- scalar or ndarray ``(exp(x) - 1)/x``, computed element-wise. See Also -------- expm1 Notes ----- .. versionadded:: 0.17.0 Examples -------- >>> import numpy as np >>> from scipy.special import exprel >>> exprel(0.01) 1.0050167084168056 >>> exprel([-0.25, -0.1, 0, 0.1, 0.25]) array([ 0.88479687, 0.95162582, 1. , 1.05170918, 1.13610167]) Compare ``exprel(5e-9)`` to the naive calculation. The exact value is ``1.00000000250000000416...``. >>> exprel(5e-9) 1.0000000025 >>> (np.exp(5e-9) - 1)/5e-9 0.99999999392252903fdtrfdtr(dfn, dfd, x, out=None) F cumulative distribution function. Returns the value of the cumulative distribution function of the F-distribution, also known as Snedecor's F-distribution or the Fisher-Snedecor distribution. The F-distribution with parameters :math:`d_n` and :math:`d_d` is the distribution of the random variable, .. math:: X = \frac{U_n/d_n}{U_d/d_d}, where :math:`U_n` and :math:`U_d` are random variables distributed :math:`\chi^2`, with :math:`d_n` and :math:`d_d` degrees of freedom, respectively. Parameters ---------- dfn : array_like First parameter (positive float). dfd : array_like Second parameter (positive float). x : array_like Argument (nonnegative float). out : ndarray, optional Optional output array for the function values Returns ------- y : scalar or ndarray The CDF of the F-distribution with parameters `dfn` and `dfd` at `x`. See Also -------- fdtrc : F distribution survival function fdtri : F distribution inverse cumulative distribution scipy.stats.f : F distribution Notes ----- The regularized incomplete beta function is used, according to the formula, .. math:: F(d_n, d_d; x) = I_{xd_n/(d_d + xd_n)}(d_n/2, d_d/2). Wrapper for the Cephes [1]_ routine `fdtr`. The F distribution is also available as `scipy.stats.f`. Calling `fdtr` directly can improve performance compared to the ``cdf`` method of `scipy.stats.f` (see last example below). References ---------- .. [1] Cephes Mathematical Functions Library, http://www.netlib.org/cephes/ Examples -------- Calculate the function for ``dfn=1`` and ``dfd=2`` at ``x=1``. >>> import numpy as np >>> from scipy.special import fdtr >>> fdtr(1, 2, 1) 0.5773502691896258 Calculate the function at several points by providing a NumPy array for `x`. >>> x = np.array([0.5, 2., 3.]) >>> fdtr(1, 2, x) array([0.4472136 , 0.70710678, 0.77459667]) Plot the function for several parameter sets. >>> import matplotlib.pyplot as plt >>> dfn_parameters = [1, 5, 10, 50] >>> dfd_parameters = [1, 1, 2, 3] >>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot'] >>> parameters_list = list(zip(dfn_parameters, dfd_parameters, ... linestyles)) >>> x = np.linspace(0, 30, 1000) >>> fig, ax = plt.subplots() >>> for parameter_set in parameters_list: ... dfn, dfd, style = parameter_set ... fdtr_vals = fdtr(dfn, dfd, x) ... ax.plot(x, fdtr_vals, label=rf"$d_n={dfn},\, d_d={dfd}$", ... ls=style) >>> ax.legend() >>> ax.set_xlabel("$x$") >>> ax.set_title("F distribution cumulative distribution function") >>> plt.show() The F distribution is also available as `scipy.stats.f`. Using `fdtr` directly can be much faster than calling the ``cdf`` method of `scipy.stats.f`, especially for small arrays or individual values. To get the same results one must use the following parametrization: ``stats.f(dfn, dfd).cdf(x)=fdtr(dfn, dfd, x)``. >>> from scipy.stats import f >>> dfn, dfd = 1, 2 >>> x = 1 >>> fdtr_res = fdtr(dfn, dfd, x) # this will often be faster than below >>> f_dist_res = f(dfn, dfd).cdf(x) >>> fdtr_res == f_dist_res # test that results are equal Truefdtrcfdtrc(dfn, dfd, x, out=None) F survival function. Returns the complemented F-distribution function (the integral of the density from `x` to infinity). Parameters ---------- dfn : array_like First parameter (positive float). dfd : array_like Second parameter (positive float). x : array_like Argument (nonnegative float). out : ndarray, optional Optional output array for the function values Returns ------- y : scalar or ndarray The complemented F-distribution function with parameters `dfn` and `dfd` at `x`. See Also -------- fdtr : F distribution cumulative distribution function fdtri : F distribution inverse cumulative distribution function scipy.stats.f : F distribution Notes ----- The regularized incomplete beta function is used, according to the formula, .. math:: F(d_n, d_d; x) = I_{d_d/(d_d + xd_n)}(d_d/2, d_n/2). Wrapper for the Cephes [1]_ routine `fdtrc`. The F distribution is also available as `scipy.stats.f`. Calling `fdtrc` directly can improve performance compared to the ``sf`` method of `scipy.stats.f` (see last example below). References ---------- .. [1] Cephes Mathematical Functions Library, http://www.netlib.org/cephes/ Examples -------- Calculate the function for ``dfn=1`` and ``dfd=2`` at ``x=1``. >>> import numpy as np >>> from scipy.special import fdtrc >>> fdtrc(1, 2, 1) 0.42264973081037427 Calculate the function at several points by providing a NumPy array for `x`. >>> x = np.array([0.5, 2., 3.]) >>> fdtrc(1, 2, x) array([0.5527864 , 0.29289322, 0.22540333]) Plot the function for several parameter sets. >>> import matplotlib.pyplot as plt >>> dfn_parameters = [1, 5, 10, 50] >>> dfd_parameters = [1, 1, 2, 3] >>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot'] >>> parameters_list = list(zip(dfn_parameters, dfd_parameters, ... linestyles)) >>> x = np.linspace(0, 30, 1000) >>> fig, ax = plt.subplots() >>> for parameter_set in parameters_list: ... dfn, dfd, style = parameter_set ... fdtrc_vals = fdtrc(dfn, dfd, x) ... ax.plot(x, fdtrc_vals, label=rf"$d_n={dfn},\, d_d={dfd}$", ... ls=style) >>> ax.legend() >>> ax.set_xlabel("$x$") >>> ax.set_title("F distribution survival function") >>> plt.show() The F distribution is also available as `scipy.stats.f`. Using `fdtrc` directly can be much faster than calling the ``sf`` method of `scipy.stats.f`, especially for small arrays or individual values. To get the same results one must use the following parametrization: ``stats.f(dfn, dfd).sf(x)=fdtrc(dfn, dfd, x)``. >>> from scipy.stats import f >>> dfn, dfd = 1, 2 >>> x = 1 >>> fdtrc_res = fdtrc(dfn, dfd, x) # this will often be faster than below >>> f_dist_res = f(dfn, dfd).sf(x) >>> f_dist_res == fdtrc_res # test that results are equal Truefdtrifdtri(dfn, dfd, p, out=None) The `p`-th quantile of the F-distribution. This function is the inverse of the F-distribution CDF, `fdtr`, returning the `x` such that `fdtr(dfn, dfd, x) = p`. Parameters ---------- dfn : array_like First parameter (positive float). dfd : array_like Second parameter (positive float). p : array_like Cumulative probability, in [0, 1]. out : ndarray, optional Optional output array for the function values Returns ------- x : scalar or ndarray The quantile corresponding to `p`. See Also -------- fdtr : F distribution cumulative distribution function fdtrc : F distribution survival function scipy.stats.f : F distribution Notes ----- The computation is carried out using the relation to the inverse regularized beta function, :math:`I^{-1}_x(a, b)`. Let :math:`z = I^{-1}_p(d_d/2, d_n/2).` Then, .. math:: x = \frac{d_d (1 - z)}{d_n z}. If `p` is such that :math:`x < 0.5`, the following relation is used instead for improved stability: let :math:`z' = I^{-1}_{1 - p}(d_n/2, d_d/2).` Then, .. math:: x = \frac{d_d z'}{d_n (1 - z')}. Wrapper for the Cephes [1]_ routine `fdtri`. The F distribution is also available as `scipy.stats.f`. Calling `fdtri` directly can improve performance compared to the ``ppf`` method of `scipy.stats.f` (see last example below). References ---------- .. [1] Cephes Mathematical Functions Library, http://www.netlib.org/cephes/ Examples -------- `fdtri` represents the inverse of the F distribution CDF which is available as `fdtr`. Here, we calculate the CDF for ``df1=1``, ``df2=2`` at ``x=3``. `fdtri` then returns ``3`` given the same values for `df1`, `df2` and the computed CDF value. >>> import numpy as np >>> from scipy.special import fdtri, fdtr >>> df1, df2 = 1, 2 >>> x = 3 >>> cdf_value = fdtr(df1, df2, x) >>> fdtri(df1, df2, cdf_value) 3.000000000000006 Calculate the function at several points by providing a NumPy array for `x`. >>> x = np.array([0.1, 0.4, 0.7]) >>> fdtri(1, 2, x) array([0.02020202, 0.38095238, 1.92156863]) Plot the function for several parameter sets. >>> import matplotlib.pyplot as plt >>> dfn_parameters = [50, 10, 1, 50] >>> dfd_parameters = [0.5, 1, 1, 5] >>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot'] >>> parameters_list = list(zip(dfn_parameters, dfd_parameters, ... linestyles)) >>> x = np.linspace(0, 1, 1000) >>> fig, ax = plt.subplots() >>> for parameter_set in parameters_list: ... dfn, dfd, style = parameter_set ... fdtri_vals = fdtri(dfn, dfd, x) ... ax.plot(x, fdtri_vals, label=rf"$d_n={dfn},\, d_d={dfd}$", ... ls=style) >>> ax.legend() >>> ax.set_xlabel("$x$") >>> title = "F distribution inverse cumulative distribution function" >>> ax.set_title(title) >>> ax.set_ylim(0, 30) >>> plt.show() The F distribution is also available as `scipy.stats.f`. Using `fdtri` directly can be much faster than calling the ``ppf`` method of `scipy.stats.f`, especially for small arrays or individual values. To get the same results one must use the following parametrization: ``stats.f(dfn, dfd).ppf(x)=fdtri(dfn, dfd, x)``. >>> from scipy.stats import f >>> dfn, dfd = 1, 2 >>> x = 0.7 >>> fdtri_res = fdtri(dfn, dfd, x) # this will often be faster than below >>> f_dist_res = f(dfn, dfd).ppf(x) >>> f_dist_res == fdtri_res # test that results are equal Truefdtridfdfdtridfd(dfn, p, x, out=None) Inverse to `fdtr` vs dfd Finds the F density argument dfd such that ``fdtr(dfn, dfd, x) == p``. Parameters ---------- dfn : array_like First parameter (positive float). p : array_like Cumulative probability, in [0, 1]. x : array_like Argument (nonnegative float). out : ndarray, optional Optional output array for the function values Returns ------- dfd : scalar or ndarray `dfd` such that ``fdtr(dfn, dfd, x) == p``. See Also -------- fdtr : F distribution cumulative distribution function fdtrc : F distribution survival function fdtri : F distribution quantile function scipy.stats.f : F distribution Examples -------- Compute the F distribution cumulative distribution function for one parameter set. >>> from scipy.special import fdtridfd, fdtr >>> dfn, dfd, x = 10, 5, 2 >>> cdf_value = fdtr(dfn, dfd, x) >>> cdf_value 0.7700248806501017 Verify that `fdtridfd` recovers the original value for `dfd`: >>> fdtridfd(dfn, cdf_value, x) 5.0fresnelfresnel(z, out=None) Fresnel integrals. The Fresnel integrals are defined as .. math:: S(z) &= \int_0^z \sin(\pi t^2 /2) dt \\ C(z) &= \int_0^z \cos(\pi t^2 /2) dt. See [dlmf]_ for details. Parameters ---------- z : array_like Real or complex valued argument out : 2-tuple of ndarrays, optional Optional output arrays for the function results Returns ------- S, C : 2-tuple of scalar or ndarray Values of the Fresnel integrals See Also -------- fresnel_zeros : zeros of the Fresnel integrals References ---------- .. [dlmf] NIST Digital Library of Mathematical Functions https://dlmf.nist.gov/7.2#iii Examples -------- >>> import numpy as np >>> import scipy.special as sc As z goes to infinity along the real axis, S and C converge to 0.5. >>> S, C = sc.fresnel([0.1, 1, 10, 100, np.inf]) >>> S array([0.00052359, 0.43825915, 0.46816998, 0.4968169 , 0.5 ]) >>> C array([0.09999753, 0.7798934 , 0.49989869, 0.4999999 , 0.5 ]) They are related to the error function `erf`. >>> z = np.array([1, 2, 3, 4]) >>> zeta = 0.5 * np.sqrt(np.pi) * (1 - 1j) * z >>> S, C = sc.fresnel(z) >>> C + 1j*S array([0.7798934 +0.43825915j, 0.48825341+0.34341568j, 0.60572079+0.496313j , 0.49842603+0.42051575j]) >>> 0.5 * (1 + 1j) * sc.erf(zeta) array([0.7798934 +0.43825915j, 0.48825341+0.34341568j, 0.60572079+0.496313j , 0.49842603+0.42051575j])gamma(z, out=None) gamma function. The gamma function is defined as .. math:: \Gamma(z) = \int_0^\infty t^{z-1} e^{-t} dt for :math:`\Re(z) > 0` and is extended to the rest of the complex plane by analytic continuation. See [dlmf]_ for more details. Parameters ---------- z : array_like Real or complex valued argument out : ndarray, optional Optional output array for the function values Returns ------- scalar or ndarray Values of the gamma function Notes ----- The gamma function is often referred to as the generalized factorial since :math:`\Gamma(n + 1) = n!` for natural numbers :math:`n`. More generally it satisfies the recurrence relation :math:`\Gamma(z + 1) = z \cdot \Gamma(z)` for complex :math:`z`, which, combined with the fact that :math:`\Gamma(1) = 1`, implies the above identity for :math:`z = n`. References ---------- .. [dlmf] NIST Digital Library of Mathematical Functions https://dlmf.nist.gov/5.2#E1 Examples -------- >>> import numpy as np >>> from scipy.special import gamma, factorial >>> gamma([0, 0.5, 1, 5]) array([ inf, 1.77245385, 1. , 24. ]) >>> z = 2.5 + 1j >>> gamma(z) (0.77476210455108352+0.70763120437959293j) >>> gamma(z+1), z*gamma(z) # Recurrence property ((1.2292740569981171+2.5438401155000685j), (1.2292740569981158+2.5438401155000658j)) >>> gamma(0.5)**2 # gamma(0.5) = sqrt(pi) 3.1415926535897927 Plot gamma(x) for real x >>> x = np.linspace(-3.5, 5.5, 2251) >>> y = gamma(x) >>> import matplotlib.pyplot as plt >>> plt.plot(x, y, 'b', alpha=0.6, label='gamma(x)') >>> k = np.arange(1, 7) >>> plt.plot(k, factorial(k-1), 'k*', alpha=0.6, ... label='(x-1)!, x = 1, 2, ...') >>> plt.xlim(-3.5, 5.5) >>> plt.ylim(-10, 25) >>> plt.grid() >>> plt.xlabel('x') >>> plt.legend(loc='lower right') >>> plt.show()gammaincgammainc(a, x, out=None) Regularized lower incomplete gamma function. It is defined as .. math:: P(a, x) = \frac{1}{\Gamma(a)} \int_0^x t^{a - 1}e^{-t} dt for :math:`a > 0` and :math:`x \geq 0`. See [dlmf]_ for details. Parameters ---------- a : array_like Positive parameter x : array_like Nonnegative argument out : ndarray, optional Optional output array for the function values Returns ------- scalar or ndarray Values of the lower incomplete gamma function Notes ----- The function satisfies the relation ``gammainc(a, x) + gammaincc(a, x) = 1`` where `gammaincc` is the regularized upper incomplete gamma function. The implementation largely follows that of [boost]_. See also -------- gammaincc : regularized upper incomplete gamma function gammaincinv : inverse of the regularized lower incomplete gamma function gammainccinv : inverse of the regularized upper incomplete gamma function References ---------- .. [dlmf] NIST Digital Library of Mathematical functions https://dlmf.nist.gov/8.2#E4 .. [boost] Maddock et. al., "Incomplete Gamma Functions", https://www.boost.org/doc/libs/1_61_0/libs/math/doc/html/math_toolkit/sf_gamma/igamma.html Examples -------- >>> import scipy.special as sc It is the CDF of the gamma distribution, so it starts at 0 and monotonically increases to 1. >>> sc.gammainc(0.5, [0, 1, 10, 100]) array([0. , 0.84270079, 0.99999226, 1. ]) It is equal to one minus the upper incomplete gamma function. >>> a, x = 0.5, 0.4 >>> sc.gammainc(a, x) 0.6289066304773024 >>> 1 - sc.gammaincc(a, x) 0.6289066304773024gammainccgammaincc(a, x, out=None) Regularized upper incomplete gamma function. It is defined as .. math:: Q(a, x) = \frac{1}{\Gamma(a)} \int_x^\infty t^{a - 1}e^{-t} dt for :math:`a > 0` and :math:`x \geq 0`. See [dlmf]_ for details. Parameters ---------- a : array_like Positive parameter x : array_like Nonnegative argument out : ndarray, optional Optional output array for the function values Returns ------- scalar or ndarray Values of the upper incomplete gamma function Notes ----- The function satisfies the relation ``gammainc(a, x) + gammaincc(a, x) = 1`` where `gammainc` is the regularized lower incomplete gamma function. The implementation largely follows that of [boost]_. See also -------- gammainc : regularized lower incomplete gamma function gammaincinv : inverse of the regularized lower incomplete gamma function gammainccinv : inverse of the regularized upper incomplete gamma function References ---------- .. [dlmf] NIST Digital Library of Mathematical functions https://dlmf.nist.gov/8.2#E4 .. [boost] Maddock et. al., "Incomplete Gamma Functions", https://www.boost.org/doc/libs/1_61_0/libs/math/doc/html/math_toolkit/sf_gamma/igamma.html Examples -------- >>> import scipy.special as sc It is the survival function of the gamma distribution, so it starts at 1 and monotonically decreases to 0. >>> sc.gammaincc(0.5, [0, 1, 10, 100, 1000]) array([1.00000000e+00, 1.57299207e-01, 7.74421643e-06, 2.08848758e-45, 0.00000000e+00]) It is equal to one minus the lower incomplete gamma function. >>> a, x = 0.5, 0.4 >>> sc.gammaincc(a, x) 0.37109336952269756 >>> 1 - sc.gammainc(a, x) 0.37109336952269756gammainccinvgammainccinv(a, y, out=None) Inverse of the regularized upper incomplete gamma function. Given an input :math:`y` between 0 and 1, returns :math:`x` such that :math:`y = Q(a, x)`. Here :math:`Q` is the regularized upper incomplete gamma function; see `gammaincc`. This is well-defined because the upper incomplete gamma function is monotonic as can be seen from its definition in [dlmf]_. Parameters ---------- a : array_like Positive parameter y : array_like Argument between 0 and 1, inclusive out : ndarray, optional Optional output array for the function values Returns ------- scalar or ndarray Values of the inverse of the upper incomplete gamma function See Also -------- gammaincc : regularized upper incomplete gamma function gammainc : regularized lower incomplete gamma function gammaincinv : inverse of the regularized lower incomplete gamma function References ---------- .. [dlmf] NIST Digital Library of Mathematical Functions https://dlmf.nist.gov/8.2#E4 Examples -------- >>> import scipy.special as sc It starts at infinity and monotonically decreases to 0. >>> sc.gammainccinv(0.5, [0, 0.1, 0.5, 1]) array([ inf, 1.35277173, 0.22746821, 0. ]) It inverts the upper incomplete gamma function. >>> a, x = 0.5, [0, 0.1, 0.5, 1] >>> sc.gammaincc(a, sc.gammainccinv(a, x)) array([0. , 0.1, 0.5, 1. ]) >>> a, x = 0.5, [0, 10, 50] >>> sc.gammainccinv(a, sc.gammaincc(a, x)) array([ 0., 10., 50.])gammaincinvgammaincinv(a, y, out=None) Inverse to the regularized lower incomplete gamma function. Given an input :math:`y` between 0 and 1, returns :math:`x` such that :math:`y = P(a, x)`. Here :math:`P` is the regularized lower incomplete gamma function; see `gammainc`. This is well-defined because the lower incomplete gamma function is monotonic as can be seen from its definition in [dlmf]_. Parameters ---------- a : array_like Positive parameter y : array_like Parameter between 0 and 1, inclusive out : ndarray, optional Optional output array for the function values Returns ------- scalar or ndarray Values of the inverse of the lower incomplete gamma function See Also -------- gammainc : regularized lower incomplete gamma function gammaincc : regularized upper incomplete gamma function gammainccinv : inverse of the regularized upper incomplete gamma function References ---------- .. [dlmf] NIST Digital Library of Mathematical Functions https://dlmf.nist.gov/8.2#E4 Examples -------- >>> import scipy.special as sc It starts at 0 and monotonically increases to infinity. >>> sc.gammaincinv(0.5, [0, 0.1 ,0.5, 1]) array([0. , 0.00789539, 0.22746821, inf]) It inverts the lower incomplete gamma function. >>> a, x = 0.5, [0, 0.1, 0.5, 1] >>> sc.gammainc(a, sc.gammaincinv(a, x)) array([0. , 0.1, 0.5, 1. ]) >>> a, x = 0.5, [0, 10, 25] >>> sc.gammaincinv(a, sc.gammainc(a, x)) array([ 0. , 10. , 25.00001465])gammalngammaln(x, out=None) Logarithm of the absolute value of the gamma function. Defined as .. math:: \ln(\lvert\Gamma(x)\rvert) where :math:`\Gamma` is the gamma function. For more details on the gamma function, see [dlmf]_. Parameters ---------- x : array_like Real argument out : ndarray, optional Optional output array for the function results Returns ------- scalar or ndarray Values of the log of the absolute value of gamma See Also -------- gammasgn : sign of the gamma function loggamma : principal branch of the logarithm of the gamma function Notes ----- It is the same function as the Python standard library function :func:`math.lgamma`. When used in conjunction with `gammasgn`, this function is useful for working in logspace on the real axis without having to deal with complex numbers via the relation ``exp(gammaln(x)) = gammasgn(x) * gamma(x)``. For complex-valued log-gamma, use `loggamma` instead of `gammaln`. References ---------- .. [dlmf] NIST Digital Library of Mathematical Functions https://dlmf.nist.gov/5 Examples -------- >>> import numpy as np >>> import scipy.special as sc It has two positive zeros. >>> sc.gammaln([1, 2]) array([0., 0.]) It has poles at nonpositive integers. >>> sc.gammaln([0, -1, -2, -3, -4]) array([inf, inf, inf, inf, inf]) It asymptotically approaches ``x * log(x)`` (Stirling's formula). >>> x = np.array([1e10, 1e20, 1e40, 1e80]) >>> sc.gammaln(x) array([2.20258509e+11, 4.50517019e+21, 9.11034037e+41, 1.83206807e+82]) >>> x * np.log(x) array([2.30258509e+11, 4.60517019e+21, 9.21034037e+41, 1.84206807e+82])gammasgngammasgn(x, out=None) Sign of the gamma function. It is defined as .. math:: \text{gammasgn}(x) = \begin{cases} +1 & \Gamma(x) > 0 \\ -1 & \Gamma(x) < 0 \end{cases} where :math:`\Gamma` is the gamma function; see `gamma`. This definition is complete since the gamma function is never zero; see the discussion after [dlmf]_. Parameters ---------- x : array_like Real argument out : ndarray, optional Optional output array for the function values Returns ------- scalar or ndarray Sign of the gamma function Notes ----- The gamma function can be computed as ``gammasgn(x) * np.exp(gammaln(x))``. See Also -------- gamma : the gamma function gammaln : log of the absolute value of the gamma function loggamma : analytic continuation of the log of the gamma function References ---------- .. [dlmf] NIST Digital Library of Mathematical Functions https://dlmf.nist.gov/5.2#E1 Examples -------- >>> import numpy as np >>> import scipy.special as sc It is 1 for `x > 0`. >>> sc.gammasgn([1, 2, 3, 4]) array([1., 1., 1., 1.]) It alternates between -1 and 1 for negative integers. >>> sc.gammasgn([-0.5, -1.5, -2.5, -3.5]) array([-1., 1., -1., 1.]) It can be used to compute the gamma function. >>> x = [1.5, 0.5, -0.5, -1.5] >>> sc.gammasgn(x) * np.exp(sc.gammaln(x)) array([ 0.88622693, 1.77245385, -3.5449077 , 2.3632718 ]) >>> sc.gamma(x) array([ 0.88622693, 1.77245385, -3.5449077 , 2.3632718 ])gdtrgdtr(a, b, x, out=None) Gamma distribution cumulative distribution function. Returns the integral from zero to `x` of the gamma probability density function, .. math:: F = \int_0^x \frac{a^b}{\Gamma(b)} t^{b-1} e^{-at}\,dt, where :math:`\Gamma` is the gamma function. Parameters ---------- a : array_like The rate parameter of the gamma distribution, sometimes denoted :math:`\beta` (float). It is also the reciprocal of the scale parameter :math:`\theta`. b : array_like The shape parameter of the gamma distribution, sometimes denoted :math:`\alpha` (float). x : array_like The quantile (upper limit of integration; float). out : ndarray, optional Optional output array for the function values See also -------- gdtrc : 1 - CDF of the gamma distribution. scipy.stats.gamma: Gamma distribution Returns ------- F : scalar or ndarray The CDF of the gamma distribution with parameters `a` and `b` evaluated at `x`. Notes ----- The evaluation is carried out using the relation to the incomplete gamma integral (regularized gamma function). Wrapper for the Cephes [1]_ routine `gdtr`. Calling `gdtr` directly can improve performance compared to the ``cdf`` method of `scipy.stats.gamma` (see last example below). References ---------- .. [1] Cephes Mathematical Functions Library, http://www.netlib.org/cephes/ Examples -------- Compute the function for ``a=1``, ``b=2`` at ``x=5``. >>> import numpy as np >>> from scipy.special import gdtr >>> import matplotlib.pyplot as plt >>> gdtr(1., 2., 5.) 0.9595723180054873 Compute the function for ``a=1`` and ``b=2`` at several points by providing a NumPy array for `x`. >>> xvalues = np.array([1., 2., 3., 4]) >>> gdtr(1., 1., xvalues) array([0.63212056, 0.86466472, 0.95021293, 0.98168436]) `gdtr` can evaluate different parameter sets by providing arrays with broadcasting compatible shapes for `a`, `b` and `x`. Here we compute the function for three different `a` at four positions `x` and ``b=3``, resulting in a 3x4 array. >>> a = np.array([[0.5], [1.5], [2.5]]) >>> x = np.array([1., 2., 3., 4]) >>> a.shape, x.shape ((3, 1), (4,)) >>> gdtr(a, 3., x) array([[0.01438768, 0.0803014 , 0.19115317, 0.32332358], [0.19115317, 0.57680992, 0.82642193, 0.9380312 ], [0.45618688, 0.87534798, 0.97974328, 0.9972306 ]]) Plot the function for four different parameter sets. >>> a_parameters = [0.3, 1, 2, 6] >>> b_parameters = [2, 10, 15, 20] >>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot'] >>> parameters_list = list(zip(a_parameters, b_parameters, linestyles)) >>> x = np.linspace(0, 30, 1000) >>> fig, ax = plt.subplots() >>> for parameter_set in parameters_list: ... a, b, style = parameter_set ... gdtr_vals = gdtr(a, b, x) ... ax.plot(x, gdtr_vals, label=f"$a= {a},\, b={b}$", ls=style) >>> ax.legend() >>> ax.set_xlabel("$x$") >>> ax.set_title("Gamma distribution cumulative distribution function") >>> plt.show() The gamma distribution is also available as `scipy.stats.gamma`. Using `gdtr` directly can be much faster than calling the ``cdf`` method of `scipy.stats.gamma`, especially for small arrays or individual values. To get the same results one must use the following parametrization: ``stats.gamma(b, scale=1/a).cdf(x)=gdtr(a, b, x)``. >>> from scipy.stats import gamma >>> a = 2. >>> b = 3 >>> x = 1. >>> gdtr_result = gdtr(a, b, x) # this will often be faster than below >>> gamma_dist_result = gamma(b, scale=1/a).cdf(x) >>> gdtr_result == gamma_dist_result # test that results are equal Truegdtrcgdtrc(a, b, x, out=None) Gamma distribution survival function. Integral from `x` to infinity of the gamma probability density function, .. math:: F = \int_x^\infty \frac{a^b}{\Gamma(b)} t^{b-1} e^{-at}\,dt, where :math:`\Gamma` is the gamma function. Parameters ---------- a : array_like The rate parameter of the gamma distribution, sometimes denoted :math:`\beta` (float). It is also the reciprocal of the scale parameter :math:`\theta`. b : array_like The shape parameter of the gamma distribution, sometimes denoted :math:`\alpha` (float). x : array_like The quantile (lower limit of integration; float). out : ndarray, optional Optional output array for the function values Returns ------- F : scalar or ndarray The survival function of the gamma distribution with parameters `a` and `b` evaluated at `x`. See Also -------- gdtr: Gamma distribution cumulative distribution function scipy.stats.gamma: Gamma distribution gdtrix Notes ----- The evaluation is carried out using the relation to the incomplete gamma integral (regularized gamma function). Wrapper for the Cephes [1]_ routine `gdtrc`. Calling `gdtrc` directly can improve performance compared to the ``sf`` method of `scipy.stats.gamma` (see last example below). References ---------- .. [1] Cephes Mathematical Functions Library, http://www.netlib.org/cephes/ Examples -------- Compute the function for ``a=1`` and ``b=2`` at ``x=5``. >>> import numpy as np >>> from scipy.special import gdtrc >>> import matplotlib.pyplot as plt >>> gdtrc(1., 2., 5.) 0.04042768199451279 Compute the function for ``a=1``, ``b=2`` at several points by providing a NumPy array for `x`. >>> xvalues = np.array([1., 2., 3., 4]) >>> gdtrc(1., 1., xvalues) array([0.36787944, 0.13533528, 0.04978707, 0.01831564]) `gdtrc` can evaluate different parameter sets by providing arrays with broadcasting compatible shapes for `a`, `b` and `x`. Here we compute the function for three different `a` at four positions `x` and ``b=3``, resulting in a 3x4 array. >>> a = np.array([[0.5], [1.5], [2.5]]) >>> x = np.array([1., 2., 3., 4]) >>> a.shape, x.shape ((3, 1), (4,)) >>> gdtrc(a, 3., x) array([[0.98561232, 0.9196986 , 0.80884683, 0.67667642], [0.80884683, 0.42319008, 0.17357807, 0.0619688 ], [0.54381312, 0.12465202, 0.02025672, 0.0027694 ]]) Plot the function for four different parameter sets. >>> a_parameters = [0.3, 1, 2, 6] >>> b_parameters = [2, 10, 15, 20] >>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot'] >>> parameters_list = list(zip(a_parameters, b_parameters, linestyles)) >>> x = np.linspace(0, 30, 1000) >>> fig, ax = plt.subplots() >>> for parameter_set in parameters_list: ... a, b, style = parameter_set ... gdtrc_vals = gdtrc(a, b, x) ... ax.plot(x, gdtrc_vals, label=f"$a= {a},\, b={b}$", ls=style) >>> ax.legend() >>> ax.set_xlabel("$x$") >>> ax.set_title("Gamma distribution survival function") >>> plt.show() The gamma distribution is also available as `scipy.stats.gamma`. Using `gdtrc` directly can be much faster than calling the ``sf`` method of `scipy.stats.gamma`, especially for small arrays or individual values. To get the same results one must use the following parametrization: ``stats.gamma(b, scale=1/a).sf(x)=gdtrc(a, b, x)``. >>> from scipy.stats import gamma >>> a = 2 >>> b = 3 >>> x = 1. >>> gdtrc_result = gdtrc(a, b, x) # this will often be faster than below >>> gamma_dist_result = gamma(b, scale=1/a).sf(x) >>> gdtrc_result == gamma_dist_result # test that results are equal Truegdtriagdtria(p, b, x, out=None) Inverse of `gdtr` vs a. Returns the inverse with respect to the parameter `a` of ``p = gdtr(a, b, x)``, the cumulative distribution function of the gamma distribution. Parameters ---------- p : array_like Probability values. b : array_like `b` parameter values of `gdtr(a, b, x)`. `b` is the "shape" parameter of the gamma distribution. x : array_like Nonnegative real values, from the domain of the gamma distribution. out : ndarray, optional If a fourth argument is given, it must be a numpy.ndarray whose size matches the broadcast result of `a`, `b` and `x`. `out` is then the array returned by the function. Returns ------- a : scalar or ndarray Values of the `a` parameter such that `p = gdtr(a, b, x)`. `1/a` is the "scale" parameter of the gamma distribution. See Also -------- gdtr : CDF of the gamma distribution. gdtrib : Inverse with respect to `b` of `gdtr(a, b, x)`. gdtrix : Inverse with respect to `x` of `gdtr(a, b, x)`. Notes ----- Wrapper for the CDFLIB [1]_ Fortran routine `cdfgam`. The cumulative distribution function `p` is computed using a routine by DiDinato and Morris [2]_. Computation of `a` involves a search for a value that produces the desired value of `p`. The search relies on the monotonicity of `p` with `a`. References ---------- .. [1] Barry Brown, James Lovato, and Kathy Russell, CDFLIB: Library of Fortran Routines for Cumulative Distribution Functions, Inverses, and Other Parameters. .. [2] DiDinato, A. R. and Morris, A. H., Computation of the incomplete gamma function ratios and their inverse. ACM Trans. Math. Softw. 12 (1986), 377-393. Examples -------- First evaluate `gdtr`. >>> from scipy.special import gdtr, gdtria >>> p = gdtr(1.2, 3.4, 5.6) >>> print(p) 0.94378087442 Verify the inverse. >>> gdtria(p, 3.4, 5.6) 1.2gdtribgdtrib(a, p, x, out=None) Inverse of `gdtr` vs b. Returns the inverse with respect to the parameter `b` of ``p = gdtr(a, b, x)``, the cumulative distribution function of the gamma distribution. Parameters ---------- a : array_like `a` parameter values of `gdtr(a, b, x)`. `1/a` is the "scale" parameter of the gamma distribution. p : array_like Probability values. x : array_like Nonnegative real values, from the domain of the gamma distribution. out : ndarray, optional If a fourth argument is given, it must be a numpy.ndarray whose size matches the broadcast result of `a`, `b` and `x`. `out` is then the array returned by the function. Returns ------- b : scalar or ndarray Values of the `b` parameter such that `p = gdtr(a, b, x)`. `b` is the "shape" parameter of the gamma distribution. See Also -------- gdtr : CDF of the gamma distribution. gdtria : Inverse with respect to `a` of `gdtr(a, b, x)`. gdtrix : Inverse with respect to `x` of `gdtr(a, b, x)`. Notes ----- Wrapper for the CDFLIB [1]_ Fortran routine `cdfgam`. The cumulative distribution function `p` is computed using a routine by DiDinato and Morris [2]_. Computation of `b` involves a search for a value that produces the desired value of `p`. The search relies on the monotonicity of `p` with `b`. References ---------- .. [1] Barry Brown, James Lovato, and Kathy Russell, CDFLIB: Library of Fortran Routines for Cumulative Distribution Functions, Inverses, and Other Parameters. .. [2] DiDinato, A. R. and Morris, A. H., Computation of the incomplete gamma function ratios and their inverse. ACM Trans. Math. Softw. 12 (1986), 377-393. Examples -------- First evaluate `gdtr`. >>> from scipy.special import gdtr, gdtrib >>> p = gdtr(1.2, 3.4, 5.6) >>> print(p) 0.94378087442 Verify the inverse. >>> gdtrib(1.2, p, 5.6) 3.3999999999723882gdtrixgdtrix(a, b, p, out=None) Inverse of `gdtr` vs x. Returns the inverse with respect to the parameter `x` of ``p = gdtr(a, b, x)``, the cumulative distribution function of the gamma distribution. This is also known as the pth quantile of the distribution. Parameters ---------- a : array_like `a` parameter values of `gdtr(a, b, x)`. `1/a` is the "scale" parameter of the gamma distribution. b : array_like `b` parameter values of `gdtr(a, b, x)`. `b` is the "shape" parameter of the gamma distribution. p : array_like Probability values. out : ndarray, optional If a fourth argument is given, it must be a numpy.ndarray whose size matches the broadcast result of `a`, `b` and `x`. `out` is then the array returned by the function. Returns ------- x : scalar or ndarray Values of the `x` parameter such that `p = gdtr(a, b, x)`. See Also -------- gdtr : CDF of the gamma distribution. gdtria : Inverse with respect to `a` of `gdtr(a, b, x)`. gdtrib : Inverse with respect to `b` of `gdtr(a, b, x)`. Notes ----- Wrapper for the CDFLIB [1]_ Fortran routine `cdfgam`. The cumulative distribution function `p` is computed using a routine by DiDinato and Morris [2]_. Computation of `x` involves a search for a value that produces the desired value of `p`. The search relies on the monotonicity of `p` with `x`. References ---------- .. [1] Barry Brown, James Lovato, and Kathy Russell, CDFLIB: Library of Fortran Routines for Cumulative Distribution Functions, Inverses, and Other Parameters. .. [2] DiDinato, A. R. and Morris, A. H., Computation of the incomplete gamma function ratios and their inverse. ACM Trans. Math. Softw. 12 (1986), 377-393. Examples -------- First evaluate `gdtr`. >>> from scipy.special import gdtr, gdtrix >>> p = gdtr(1.2, 3.4, 5.6) >>> print(p) 0.94378087442 Verify the inverse. >>> gdtrix(1.2, 3.4, p) 5.5999999999999996hankel1hankel1(v, z, out=None) Hankel function of the first kind Parameters ---------- v : array_like Order (float). z : array_like Argument (float or complex). out : ndarray, optional Optional output array for the function values Returns ------- scalar or ndarray Values of the Hankel function of the first kind. Notes ----- A wrapper for the AMOS [1]_ routine `zbesh`, which carries out the computation using the relation, .. math:: H^{(1)}_v(z) = \frac{2}{\imath\pi} \exp(-\imath \pi v/2) K_v(z \exp(-\imath\pi/2)) where :math:`K_v` is the modified Bessel function of the second kind. For negative orders, the relation .. math:: H^{(1)}_{-v}(z) = H^{(1)}_v(z) \exp(\imath\pi v) is used. See also -------- hankel1e : ndarray This function with leading exponential behavior stripped off. References ---------- .. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions of a Complex Argument and Nonnegative Order", http://netlib.org/amos/hankel1ehankel1e(v, z, out=None) Exponentially scaled Hankel function of the first kind Defined as:: hankel1e(v, z) = hankel1(v, z) * exp(-1j * z) Parameters ---------- v : array_like Order (float). z : array_like Argument (float or complex). out : ndarray, optional Optional output array for the function values Returns ------- scalar or ndarray Values of the exponentially scaled Hankel function. Notes ----- A wrapper for the AMOS [1]_ routine `zbesh`, which carries out the computation using the relation, .. math:: H^{(1)}_v(z) = \frac{2}{\imath\pi} \exp(-\imath \pi v/2) K_v(z \exp(-\imath\pi/2)) where :math:`K_v` is the modified Bessel function of the second kind. For negative orders, the relation .. math:: H^{(1)}_{-v}(z) = H^{(1)}_v(z) \exp(\imath\pi v) is used. References ---------- .. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions of a Complex Argument and Nonnegative Order", http://netlib.org/amos/hankel2hankel2(v, z, out=None) Hankel function of the second kind Parameters ---------- v : array_like Order (float). z : array_like Argument (float or complex). out : ndarray, optional Optional output array for the function values Returns ------- scalar or ndarray Values of the Hankel function of the second kind. Notes ----- A wrapper for the AMOS [1]_ routine `zbesh`, which carries out the computation using the relation, .. math:: H^{(2)}_v(z) = -\frac{2}{\imath\pi} \exp(\imath \pi v/2) K_v(z \exp(\imath\pi/2)) where :math:`K_v` is the modified Bessel function of the second kind. For negative orders, the relation .. math:: H^{(2)}_{-v}(z) = H^{(2)}_v(z) \exp(-\imath\pi v) is used. See also -------- hankel2e : this function with leading exponential behavior stripped off. References ---------- .. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions of a Complex Argument and Nonnegative Order", http://netlib.org/amos/hankel2ehankel2e(v, z, out=None) Exponentially scaled Hankel function of the second kind Defined as:: hankel2e(v, z) = hankel2(v, z) * exp(1j * z) Parameters ---------- v : array_like Order (float). z : array_like Argument (float or complex). out : ndarray, optional Optional output array for the function values Returns ------- scalar or ndarray Values of the exponentially scaled Hankel function of the second kind. Notes ----- A wrapper for the AMOS [1]_ routine `zbesh`, which carries out the computation using the relation, .. math:: H^{(2)}_v(z) = -\frac{2}{\imath\pi} \exp(\frac{\imath \pi v}{2}) K_v(z exp(\frac{\imath\pi}{2})) where :math:`K_v` is the modified Bessel function of the second kind. For negative orders, the relation .. math:: H^{(2)}_{-v}(z) = H^{(2)}_v(z) \exp(-\imath\pi v) is used. References ---------- .. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions of a Complex Argument and Nonnegative Order", http://netlib.org/amos/huberhuber(delta, r, out=None) Huber loss function. .. math:: \text{huber}(\delta, r) = \begin{cases} \infty & \delta < 0 \\ \frac{1}{2}r^2 & 0 \le \delta, | r | \le \delta \\ \delta ( |r| - \frac{1}{2}\delta ) & \text{otherwise} \end{cases} Parameters ---------- delta : ndarray Input array, indicating the quadratic vs. linear loss changepoint. r : ndarray Input array, possibly representing residuals. out : ndarray, optional Optional output array for the function values Returns ------- scalar or ndarray The computed Huber loss function values. See also -------- pseudo_huber : smooth approximation of this function Notes ----- `huber` is useful as a loss function in robust statistics or machine learning to reduce the influence of outliers as compared to the common squared error loss, residuals with a magnitude higher than `delta` are not squared [1]_. Typically, `r` represents residuals, the difference between a model prediction and data. Then, for :math:`|r|\leq\delta`, `huber` resembles the squared error and for :math:`|r|>\delta` the absolute error. This way, the Huber loss often achieves a fast convergence in model fitting for small residuals like the squared error loss function and still reduces the influence of outliers (:math:`|r|>\delta`) like the absolute error loss. As :math:`\delta` is the cutoff between squared and absolute error regimes, it has to be tuned carefully for each problem. `huber` is also convex, making it suitable for gradient based optimization. .. versionadded:: 0.15.0 References ---------- .. [1] Peter Huber. "Robust Estimation of a Location Parameter", 1964. Annals of Statistics. 53 (1): 73 - 101. Examples -------- Import all necessary modules. >>> import numpy as np >>> from scipy.special import huber >>> import matplotlib.pyplot as plt Compute the function for ``delta=1`` at ``r=2`` >>> huber(1., 2.) 1.5 Compute the function for different `delta` by providing a NumPy array or list for `delta`. >>> huber([1., 3., 5.], 4.) array([3.5, 7.5, 8. ]) Compute the function at different points by providing a NumPy array or list for `r`. >>> huber(2., np.array([1., 1.5, 3.])) array([0.5 , 1.125, 4. ]) The function can be calculated for different `delta` and `r` by providing arrays for both with compatible shapes for broadcasting. >>> r = np.array([1., 2.5, 8., 10.]) >>> deltas = np.array([[1.], [5.], [9.]]) >>> print(r.shape, deltas.shape) (4,) (3, 1) >>> huber(deltas, r) array([[ 0.5 , 2. , 7.5 , 9.5 ], [ 0.5 , 3.125, 27.5 , 37.5 ], [ 0.5 , 3.125, 32. , 49.5 ]]) Plot the function for different `delta`. >>> x = np.linspace(-4, 4, 500) >>> deltas = [1, 2, 3] >>> linestyles = ["dashed", "dotted", "dashdot"] >>> fig, ax = plt.subplots() >>> combined_plot_parameters = list(zip(deltas, linestyles)) >>> for delta, style in combined_plot_parameters: ... ax.plot(x, huber(delta, x), label=f"$\delta={delta}$", ls=style) >>> ax.legend(loc="upper center") >>> ax.set_xlabel("$x$") >>> ax.set_title("Huber loss function $h_{\delta}(x)$") >>> ax.set_xlim(-4, 4) >>> ax.set_ylim(0, 8) >>> plt.show()hyp0f1hyp0f1(v, z, out=None) Confluent hypergeometric limit function 0F1. Parameters ---------- v : array_like Real-valued parameter z : array_like Real- or complex-valued argument out : ndarray, optional Optional output array for the function results Returns ------- scalar or ndarray The confluent hypergeometric limit function Notes ----- This function is defined as: .. math:: _0F_1(v, z) = \sum_{k=0}^{\infty}\frac{z^k}{(v)_k k!}. It's also the limit as :math:`q \to \infty` of :math:`_1F_1(q; v; z/q)`, and satisfies the differential equation :math:`f''(z) + vf'(z) = f(z)`. See [1]_ for more information. References ---------- .. [1] Wolfram MathWorld, "Confluent Hypergeometric Limit Function", http://mathworld.wolfram.com/ConfluentHypergeometricLimitFunction.html Examples -------- >>> import numpy as np >>> import scipy.special as sc It is one when `z` is zero. >>> sc.hyp0f1(1, 0) 1.0 It is the limit of the confluent hypergeometric function as `q` goes to infinity. >>> q = np.array([1, 10, 100, 1000]) >>> v = 1 >>> z = 1 >>> sc.hyp1f1(q, v, z / q) array([2.71828183, 2.31481985, 2.28303778, 2.27992985]) >>> sc.hyp0f1(v, z) 2.2795853023360673 It is related to Bessel functions. >>> n = 1 >>> x = np.linspace(0, 1, 5) >>> sc.jv(n, x) array([0. , 0.12402598, 0.24226846, 0.3492436 , 0.44005059]) >>> (0.5 * x)**n / sc.factorial(n) * sc.hyp0f1(n + 1, -0.25 * x**2) array([0. , 0.12402598, 0.24226846, 0.3492436 , 0.44005059])hyp1f1hyp1f1(a, b, x, out=None) Confluent hypergeometric function 1F1. The confluent hypergeometric function is defined by the series .. math:: {}_1F_1(a; b; x) = \sum_{k = 0}^\infty \frac{(a)_k}{(b)_k k!} x^k. See [dlmf]_ for more details. Here :math:`(\cdot)_k` is the Pochhammer symbol; see `poch`. Parameters ---------- a, b : array_like Real parameters x : array_like Real or complex argument out : ndarray, optional Optional output array for the function results Returns ------- scalar or ndarray Values of the confluent hypergeometric function See also -------- hyperu : another confluent hypergeometric function hyp0f1 : confluent hypergeometric limit function hyp2f1 : Gaussian hypergeometric function References ---------- .. [dlmf] NIST Digital Library of Mathematical Functions https://dlmf.nist.gov/13.2#E2 Examples -------- >>> import numpy as np >>> import scipy.special as sc It is one when `x` is zero: >>> sc.hyp1f1(0.5, 0.5, 0) 1.0 It is singular when `b` is a nonpositive integer. >>> sc.hyp1f1(0.5, -1, 0) inf It is a polynomial when `a` is a nonpositive integer. >>> a, b, x = -1, 0.5, np.array([1.0, 2.0, 3.0, 4.0]) >>> sc.hyp1f1(a, b, x) array([-1., -3., -5., -7.]) >>> 1 + (a / b) * x array([-1., -3., -5., -7.]) It reduces to the exponential function when `a = b`. >>> sc.hyp1f1(2, 2, [1, 2, 3, 4]) array([ 2.71828183, 7.3890561 , 20.08553692, 54.59815003]) >>> np.exp([1, 2, 3, 4]) array([ 2.71828183, 7.3890561 , 20.08553692, 54.59815003])hyp2f1(a, b, c, z, out=None) Gauss hypergeometric function 2F1(a, b; c; z) Parameters ---------- a, b, c : array_like Arguments, should be real-valued. z : array_like Argument, real or complex. out : ndarray, optional Optional output array for the function values Returns ------- hyp2f1 : scalar or ndarray The values of the gaussian hypergeometric function. See also -------- hyp0f1 : confluent hypergeometric limit function. hyp1f1 : Kummer's (confluent hypergeometric) function. Notes ----- This function is defined for :math:`|z| < 1` as .. math:: \mathrm{hyp2f1}(a, b, c, z) = \sum_{n=0}^\infty \frac{(a)_n (b)_n}{(c)_n}\frac{z^n}{n!}, and defined on the rest of the complex z-plane by analytic continuation [1]_. Here :math:`(\cdot)_n` is the Pochhammer symbol; see `poch`. When :math:`n` is an integer the result is a polynomial of degree :math:`n`. The implementation for complex values of ``z`` is described in [2]_, except for ``z`` in the region defined by .. math:: 0.9 <= \left|z\right| < 1.1, \left|1 - z\right| >= 0.9, \mathrm{real}(z) >= 0 in which the implementation follows [4]_. References ---------- .. [1] NIST Digital Library of Mathematical Functions https://dlmf.nist.gov/15.2 .. [2] S. Zhang and J.M. Jin, "Computation of Special Functions", Wiley 1996 .. [3] Cephes Mathematical Functions Library, http://www.netlib.org/cephes/ .. [4] J.L. Lopez and N.M. Temme, "New series expansions of the Gauss hypergeometric function", Adv Comput Math 39, 349-365 (2013). https://doi.org/10.1007/s10444-012-9283-y Examples -------- >>> import numpy as np >>> import scipy.special as sc It has poles when `c` is a negative integer. >>> sc.hyp2f1(1, 1, -2, 1) inf It is a polynomial when `a` or `b` is a negative integer. >>> a, b, c = -1, 1, 1.5 >>> z = np.linspace(0, 1, 5) >>> sc.hyp2f1(a, b, c, z) array([1. , 0.83333333, 0.66666667, 0.5 , 0.33333333]) >>> 1 + a * b * z / c array([1. , 0.83333333, 0.66666667, 0.5 , 0.33333333]) It is symmetric in `a` and `b`. >>> a = np.linspace(0, 1, 5) >>> b = np.linspace(0, 1, 5) >>> sc.hyp2f1(a, b, 1, 0.5) array([1. , 1.03997334, 1.1803406 , 1.47074441, 2. ]) >>> sc.hyp2f1(b, a, 1, 0.5) array([1. , 1.03997334, 1.1803406 , 1.47074441, 2. ]) It contains many other functions as special cases. >>> z = 0.5 >>> sc.hyp2f1(1, 1, 2, z) 1.3862943611198901 >>> -np.log(1 - z) / z 1.3862943611198906 >>> sc.hyp2f1(0.5, 1, 1.5, z**2) 1.098612288668109 >>> np.log((1 + z) / (1 - z)) / (2 * z) 1.0986122886681098 >>> sc.hyp2f1(0.5, 1, 1.5, -z**2) 0.9272952180016117 >>> np.arctan(z) / z 0.9272952180016122hyperu(a, b, x, out=None) Confluent hypergeometric function U It is defined as the solution to the equation .. math:: x \frac{d^2w}{dx^2} + (b - x) \frac{dw}{dx} - aw = 0 which satisfies the property .. math:: U(a, b, x) \sim x^{-a} as :math:`x \to \infty`. See [dlmf]_ for more details. Parameters ---------- a, b : array_like Real-valued parameters x : array_like Real-valued argument out : ndarray, optional Optional output array for the function values Returns ------- scalar or ndarray Values of `U` References ---------- .. [dlmf] NIST Digital Library of Mathematics Functions https://dlmf.nist.gov/13.2#E6 Examples -------- >>> import numpy as np >>> import scipy.special as sc It has a branch cut along the negative `x` axis. >>> x = np.linspace(-0.1, -10, 5) >>> sc.hyperu(1, 1, x) array([nan, nan, nan, nan, nan]) It approaches zero as `x` goes to infinity. >>> x = np.array([1, 10, 100]) >>> sc.hyperu(1, 1, x) array([0.59634736, 0.09156333, 0.00990194]) It satisfies Kummer's transformation. >>> a, b, x = 2, 1, 1 >>> sc.hyperu(a, b, x) 0.1926947246463881 >>> x**(1 - b) * sc.hyperu(a - b + 1, 2 - b, x) 0.1926947246463881i0i0(x, out=None) Modified Bessel function of order 0. Defined as, .. math:: I_0(x) = \sum_{k=0}^\infty \frac{(x^2/4)^k}{(k!)^2} = J_0(\imath x), where :math:`J_0` is the Bessel function of the first kind of order 0. Parameters ---------- x : array_like Argument (float) out : ndarray, optional Optional output array for the function values Returns ------- I : scalar or ndarray Value of the modified Bessel function of order 0 at `x`. Notes ----- The range is partitioned into the two intervals [0, 8] and (8, infinity). Chebyshev polynomial expansions are employed in each interval. This function is a wrapper for the Cephes [1]_ routine `i0`. See also -------- iv: Modified Bessel function of any order i0e: Exponentially scaled modified Bessel function of order 0 References ---------- .. [1] Cephes Mathematical Functions Library, http://www.netlib.org/cephes/ Examples -------- Calculate the function at one point: >>> from scipy.special import i0 >>> i0(1.) 1.2660658777520082 Calculate at several points: >>> import numpy as np >>> i0(np.array([-2., 0., 3.5])) array([2.2795853 , 1. , 7.37820343]) Plot the function from -10 to 10. >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots() >>> x = np.linspace(-10., 10., 1000) >>> y = i0(x) >>> ax.plot(x, y) >>> plt.show()i0ei0e(x, out=None) Exponentially scaled modified Bessel function of order 0. Defined as:: i0e(x) = exp(-abs(x)) * i0(x). Parameters ---------- x : array_like Argument (float) out : ndarray, optional Optional output array for the function values Returns ------- I : scalar or ndarray Value of the exponentially scaled modified Bessel function of order 0 at `x`. Notes ----- The range is partitioned into the two intervals [0, 8] and (8, infinity). Chebyshev polynomial expansions are employed in each interval. The polynomial expansions used are the same as those in `i0`, but they are not multiplied by the dominant exponential factor. This function is a wrapper for the Cephes [1]_ routine `i0e`. `i0e` is useful for large arguments `x`: for these, `i0` quickly overflows. See also -------- iv: Modified Bessel function of the first kind i0: Modified Bessel function of order 0 References ---------- .. [1] Cephes Mathematical Functions Library, http://www.netlib.org/cephes/ Examples -------- In the following example `i0` returns infinity whereas `i0e` still returns a finite number. >>> from scipy.special import i0, i0e >>> i0(1000.), i0e(1000.) (inf, 0.012617240455891257) Calculate the function at several points by providing a NumPy array or list for `x`: >>> import numpy as np >>> i0e(np.array([-2., 0., 3.])) array([0.30850832, 1. , 0.24300035]) Plot the function from -10 to 10. >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots() >>> x = np.linspace(-10., 10., 1000) >>> y = i0e(x) >>> ax.plot(x, y) >>> plt.show()i1i1(x, out=None) Modified Bessel function of order 1. Defined as, .. math:: I_1(x) = \frac{1}{2}x \sum_{k=0}^\infty \frac{(x^2/4)^k}{k! (k + 1)!} = -\imath J_1(\imath x), where :math:`J_1` is the Bessel function of the first kind of order 1. Parameters ---------- x : array_like Argument (float) out : ndarray, optional Optional output array for the function values Returns ------- I : scalar or ndarray Value of the modified Bessel function of order 1 at `x`. Notes ----- The range is partitioned into the two intervals [0, 8] and (8, infinity). Chebyshev polynomial expansions are employed in each interval. This function is a wrapper for the Cephes [1]_ routine `i1`. See also -------- iv: Modified Bessel function of the first kind i1e: Exponentially scaled modified Bessel function of order 1 References ---------- .. [1] Cephes Mathematical Functions Library, http://www.netlib.org/cephes/ Examples -------- Calculate the function at one point: >>> from scipy.special import i1 >>> i1(1.) 0.5651591039924851 Calculate the function at several points: >>> import numpy as np >>> i1(np.array([-2., 0., 6.])) array([-1.59063685, 0. , 61.34193678]) Plot the function between -10 and 10. >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots() >>> x = np.linspace(-10., 10., 1000) >>> y = i1(x) >>> ax.plot(x, y) >>> plt.show()i1ei1e(x, out=None) Exponentially scaled modified Bessel function of order 1. Defined as:: i1e(x) = exp(-abs(x)) * i1(x) Parameters ---------- x : array_like Argument (float) out : ndarray, optional Optional output array for the function values Returns ------- I : scalar or ndarray Value of the exponentially scaled modified Bessel function of order 1 at `x`. Notes ----- The range is partitioned into the two intervals [0, 8] and (8, infinity). Chebyshev polynomial expansions are employed in each interval. The polynomial expansions used are the same as those in `i1`, but they are not multiplied by the dominant exponential factor. This function is a wrapper for the Cephes [1]_ routine `i1e`. `i1e` is useful for large arguments `x`: for these, `i1` quickly overflows. See also -------- iv: Modified Bessel function of the first kind i1: Modified Bessel function of order 1 References ---------- .. [1] Cephes Mathematical Functions Library, http://www.netlib.org/cephes/ Examples -------- In the following example `i1` returns infinity whereas `i1e` still returns a finite number. >>> from scipy.special import i1, i1e >>> i1(1000.), i1e(1000.) (inf, 0.01261093025692863) Calculate the function at several points by providing a NumPy array or list for `x`: >>> import numpy as np >>> i1e(np.array([-2., 0., 6.])) array([-0.21526929, 0. , 0.15205146]) Plot the function between -10 and 10. >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots() >>> x = np.linspace(-10., 10., 1000) >>> y = i1e(x) >>> ax.plot(x, y) >>> plt.show()inv_boxcoxinv_boxcox(y, lmbda, out=None) Compute the inverse of the Box-Cox transformation. Find ``x`` such that:: y = (x**lmbda - 1) / lmbda if lmbda != 0 log(x) if lmbda == 0 Parameters ---------- y : array_like Data to be transformed. lmbda : array_like Power parameter of the Box-Cox transform. out : ndarray, optional Optional output array for the function values Returns ------- x : scalar or ndarray Transformed data. Notes ----- .. versionadded:: 0.16.0 Examples -------- >>> from scipy.special import boxcox, inv_boxcox >>> y = boxcox([1, 4, 10], 2.5) >>> inv_boxcox(y, 2.5) array([1., 4., 10.])inv_boxcox1pinv_boxcox1p(y, lmbda, out=None) Compute the inverse of the Box-Cox transformation. Find ``x`` such that:: y = ((1+x)**lmbda - 1) / lmbda if lmbda != 0 log(1+x) if lmbda == 0 Parameters ---------- y : array_like Data to be transformed. lmbda : array_like Power parameter of the Box-Cox transform. out : ndarray, optional Optional output array for the function values Returns ------- x : scalar or ndarray Transformed data. Notes ----- .. versionadded:: 0.16.0 Examples -------- >>> from scipy.special import boxcox1p, inv_boxcox1p >>> y = boxcox1p([1, 4, 10], 2.5) >>> inv_boxcox1p(y, 2.5) array([1., 4., 10.])it2i0k0it2i0k0(x, out=None) Integrals related to modified Bessel functions of order 0. Computes the integrals .. math:: \int_0^x \frac{I_0(t) - 1}{t} dt \\ \int_x^\infty \frac{K_0(t)}{t} dt. Parameters ---------- x : array_like Values at which to evaluate the integrals. out : tuple of ndarrays, optional Optional output arrays for the function results. Returns ------- ii0 : scalar or ndarray The integral for `i0` ik0 : scalar or ndarray The integral for `k0` References ---------- .. [1] S. Zhang and J.M. Jin, "Computation of Special Functions", Wiley 1996 Examples -------- Evaluate the functions at one point. >>> from scipy.special import it2i0k0 >>> int_i, int_k = it2i0k0(1.) >>> int_i, int_k (0.12897944249456852, 0.2085182909001295) Evaluate the functions at several points. >>> import numpy as np >>> points = np.array([0.5, 1.5, 3.]) >>> int_i, int_k = it2i0k0(points) >>> int_i, int_k (array([0.03149527, 0.30187149, 1.50012461]), array([0.66575102, 0.0823715 , 0.00823631])) Plot the functions from 0 to 5. >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots() >>> x = np.linspace(0., 5., 1000) >>> int_i, int_k = it2i0k0(x) >>> ax.plot(x, int_i, label=r"$\int_0^x \frac{I_0(t)-1}{t}\,dt$") >>> ax.plot(x, int_k, label=r"$\int_x^{\infty} \frac{K_0(t)}{t}\,dt$") >>> ax.legend() >>> ax.set_ylim(0, 10) >>> plt.show()it2j0y0it2j0y0(x, out=None) Integrals related to Bessel functions of the first kind of order 0. Computes the integrals .. math:: \int_0^x \frac{1 - J_0(t)}{t} dt \\ \int_x^\infty \frac{Y_0(t)}{t} dt. For more on :math:`J_0` and :math:`Y_0` see `j0` and `y0`. Parameters ---------- x : array_like Values at which to evaluate the integrals. out : tuple of ndarrays, optional Optional output arrays for the function results. Returns ------- ij0 : scalar or ndarray The integral for `j0` iy0 : scalar or ndarray The integral for `y0` References ---------- .. [1] S. Zhang and J.M. Jin, "Computation of Special Functions", Wiley 1996 Examples -------- Evaluate the functions at one point. >>> from scipy.special import it2j0y0 >>> int_j, int_y = it2j0y0(1.) >>> int_j, int_y (0.12116524699506871, 0.39527290169929336) Evaluate the functions at several points. >>> import numpy as np >>> points = np.array([0.5, 1.5, 3.]) >>> int_j, int_y = it2j0y0(points) >>> int_j, int_y (array([0.03100699, 0.26227724, 0.85614669]), array([ 0.26968854, 0.29769696, -0.02987272])) Plot the functions from 0 to 10. >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots() >>> x = np.linspace(0., 10., 1000) >>> int_j, int_y = it2j0y0(x) >>> ax.plot(x, int_j, label=r"$\int_0^x \frac{1-J_0(t)}{t}\,dt$") >>> ax.plot(x, int_y, label=r"$\int_x^{\infty} \frac{Y_0(t)}{t}\,dt$") >>> ax.legend() >>> ax.set_ylim(-2.5, 2.5) >>> plt.show()it2struve0it2struve0(x, out=None) Integral related to the Struve function of order 0. Returns the integral, .. math:: \int_x^\infty \frac{H_0(t)}{t}\,dt where :math:`H_0` is the Struve function of order 0. Parameters ---------- x : array_like Lower limit of integration. out : ndarray, optional Optional output array for the function values Returns ------- I : scalar or ndarray The value of the integral. See also -------- struve Notes ----- Wrapper for a Fortran routine created by Shanjie Zhang and Jianming Jin [1]_. References ---------- .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special Functions", John Wiley and Sons, 1996. https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html Examples -------- Evaluate the function at one point. >>> import numpy as np >>> from scipy.special import it2struve0 >>> it2struve0(1.) 0.9571973506383524 Evaluate the function at several points by supplying an array for `x`. >>> points = np.array([1., 2., 3.5]) >>> it2struve0(points) array([0.95719735, 0.46909296, 0.10366042]) Plot the function from -10 to 10. >>> import matplotlib.pyplot as plt >>> x = np.linspace(-10., 10., 1000) >>> it2struve0_values = it2struve0(x) >>> fig, ax = plt.subplots() >>> ax.plot(x, it2struve0_values) >>> ax.set_xlabel(r'$x$') >>> ax.set_ylabel(r'$\int_x^{\infty}\frac{H_0(t)}{t}\,dt$') >>> plt.show()itairyitairy(x, out=None) Integrals of Airy functions Calculates the integrals of Airy functions from 0 to `x`. Parameters ---------- x : array_like Upper limit of integration (float). out : tuple of ndarray, optional Optional output arrays for the function values Returns ------- Apt : scalar or ndarray Integral of Ai(t) from 0 to x. Bpt : scalar or ndarray Integral of Bi(t) from 0 to x. Ant : scalar or ndarray Integral of Ai(-t) from 0 to x. Bnt : scalar or ndarray Integral of Bi(-t) from 0 to x. Notes ----- Wrapper for a Fortran routine created by Shanjie Zhang and Jianming Jin [1]_. References ---------- .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special Functions", John Wiley and Sons, 1996. https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html Examples -------- Compute the functions at ``x=1.``. >>> import numpy as np >>> from scipy.special import itairy >>> import matplotlib.pyplot as plt >>> apt, bpt, ant, bnt = itairy(1.) >>> apt, bpt, ant, bnt (0.23631734191710949, 0.8727691167380077, 0.46567398346706845, 0.3730050096342943) Compute the functions at several points by providing a NumPy array for `x`. >>> x = np.array([1., 1.5, 2.5, 5]) >>> apt, bpt, ant, bnt = itairy(x) >>> apt, bpt, ant, bnt (array([0.23631734, 0.28678675, 0.324638 , 0.33328759]), array([ 0.87276912, 1.62470809, 5.20906691, 321.47831857]), array([0.46567398, 0.72232876, 0.93187776, 0.7178822 ]), array([ 0.37300501, 0.35038814, -0.02812939, 0.15873094])) Plot the functions from -10 to 10. >>> x = np.linspace(-10, 10, 500) >>> apt, bpt, ant, bnt = itairy(x) >>> fig, ax = plt.subplots(figsize=(6, 5)) >>> ax.plot(x, apt, label="$\int_0^x\, Ai(t)\, dt$") >>> ax.plot(x, bpt, ls="dashed", label="$\int_0^x\, Bi(t)\, dt$") >>> ax.plot(x, ant, ls="dashdot", label="$\int_0^x\, Ai(-t)\, dt$") >>> ax.plot(x, bnt, ls="dotted", label="$\int_0^x\, Bi(-t)\, dt$") >>> ax.set_ylim(-2, 1.5) >>> ax.legend(loc="lower right") >>> plt.show()iti0k0iti0k0(x, out=None) Integrals of modified Bessel functions of order 0. Computes the integrals .. math:: \int_0^x I_0(t) dt \\ \int_0^x K_0(t) dt. For more on :math:`I_0` and :math:`K_0` see `i0` and `k0`. Parameters ---------- x : array_like Values at which to evaluate the integrals. out : tuple of ndarrays, optional Optional output arrays for the function results. Returns ------- ii0 : scalar or ndarray The integral for `i0` ik0 : scalar or ndarray The integral for `k0` References ---------- .. [1] S. Zhang and J.M. Jin, "Computation of Special Functions", Wiley 1996 Examples -------- Evaluate the functions at one point. >>> from scipy.special import iti0k0 >>> int_i, int_k = iti0k0(1.) >>> int_i, int_k (1.0865210970235892, 1.2425098486237771) Evaluate the functions at several points. >>> import numpy as np >>> points = np.array([0., 1.5, 3.]) >>> int_i, int_k = iti0k0(points) >>> int_i, int_k (array([0. , 1.80606937, 6.16096149]), array([0. , 1.39458246, 1.53994809])) Plot the functions from 0 to 5. >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots() >>> x = np.linspace(0., 5., 1000) >>> int_i, int_k = iti0k0(x) >>> ax.plot(x, int_i, label="$\int_0^x I_0(t)\,dt$") >>> ax.plot(x, int_k, label="$\int_0^x K_0(t)\,dt$") >>> ax.legend() >>> plt.show()itj0y0itj0y0(x, out=None) Integrals of Bessel functions of the first kind of order 0. Computes the integrals .. math:: \int_0^x J_0(t) dt \\ \int_0^x Y_0(t) dt. For more on :math:`J_0` and :math:`Y_0` see `j0` and `y0`. Parameters ---------- x : array_like Values at which to evaluate the integrals. out : tuple of ndarrays, optional Optional output arrays for the function results. Returns ------- ij0 : scalar or ndarray The integral of `j0` iy0 : scalar or ndarray The integral of `y0` References ---------- .. [1] S. Zhang and J.M. Jin, "Computation of Special Functions", Wiley 1996 Examples -------- Evaluate the functions at one point. >>> from scipy.special import itj0y0 >>> int_j, int_y = itj0y0(1.) >>> int_j, int_y (0.9197304100897596, -0.637069376607422) Evaluate the functions at several points. >>> import numpy as np >>> points = np.array([0., 1.5, 3.]) >>> int_j, int_y = itj0y0(points) >>> int_j, int_y (array([0. , 1.24144951, 1.38756725]), array([ 0. , -0.51175903, 0.19765826])) Plot the functions from 0 to 10. >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots() >>> x = np.linspace(0., 10., 1000) >>> int_j, int_y = itj0y0(x) >>> ax.plot(x, int_j, label="$\int_0^x J_0(t)\,dt$") >>> ax.plot(x, int_y, label="$\int_0^x Y_0(t)\,dt$") >>> ax.legend() >>> plt.show()itmodstruve0itmodstruve0(x, out=None) Integral of the modified Struve function of order 0. .. math:: I = \int_0^x L_0(t)\,dt Parameters ---------- x : array_like Upper limit of integration (float). out : ndarray, optional Optional output array for the function values Returns ------- I : scalar or ndarray The integral of :math:`L_0` from 0 to `x`. Notes ----- Wrapper for a Fortran routine created by Shanjie Zhang and Jianming Jin [1]_. See Also -------- modstruve: Modified Struve function which is integrated by this function References ---------- .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special Functions", John Wiley and Sons, 1996. https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html Examples -------- Evaluate the function at one point. >>> import numpy as np >>> from scipy.special import itmodstruve0 >>> itmodstruve0(1.) 0.3364726286440384 Evaluate the function at several points by supplying an array for `x`. >>> points = np.array([1., 2., 3.5]) >>> itmodstruve0(points) array([0.33647263, 1.588285 , 7.60382578]) Plot the function from -10 to 10. >>> import matplotlib.pyplot as plt >>> x = np.linspace(-10., 10., 1000) >>> itmodstruve0_values = itmodstruve0(x) >>> fig, ax = plt.subplots() >>> ax.plot(x, itmodstruve0_values) >>> ax.set_xlabel(r'$x$') >>> ax.set_ylabel(r'$\int_0^xL_0(t)\,dt$') >>> plt.show()itstruve0itstruve0(x, out=None) Integral of the Struve function of order 0. .. math:: I = \int_0^x H_0(t)\,dt Parameters ---------- x : array_like Upper limit of integration (float). out : ndarray, optional Optional output array for the function values Returns ------- I : scalar or ndarray The integral of :math:`H_0` from 0 to `x`. See also -------- struve: Function which is integrated by this function Notes ----- Wrapper for a Fortran routine created by Shanjie Zhang and Jianming Jin [1]_. References ---------- .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special Functions", John Wiley and Sons, 1996. https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html Examples -------- Evaluate the function at one point. >>> import numpy as np >>> from scipy.special import itstruve0 >>> itstruve0(1.) 0.30109042670805547 Evaluate the function at several points by supplying an array for `x`. >>> points = np.array([1., 2., 3.5]) >>> itstruve0(points) array([0.30109043, 1.01870116, 1.96804581]) Plot the function from -20 to 20. >>> import matplotlib.pyplot as plt >>> x = np.linspace(-20., 20., 1000) >>> istruve0_values = itstruve0(x) >>> fig, ax = plt.subplots() >>> ax.plot(x, istruve0_values) >>> ax.set_xlabel(r'$x$') >>> ax.set_ylabel(r'$\int_0^{x}H_0(t)\,dt$') >>> plt.show()iviv(v, z, out=None) Modified Bessel function of the first kind of real order. Parameters ---------- v : array_like Order. If `z` is of real type and negative, `v` must be integer valued. z : array_like of float or complex Argument. out : ndarray, optional Optional output array for the function values Returns ------- scalar or ndarray Values of the modified Bessel function. Notes ----- For real `z` and :math:`v \in [-50, 50]`, the evaluation is carried out using Temme's method [1]_. For larger orders, uniform asymptotic expansions are applied. For complex `z` and positive `v`, the AMOS [2]_ `zbesi` routine is called. It uses a power series for small `z`, the asymptotic expansion for large `abs(z)`, the Miller algorithm normalized by the Wronskian and a Neumann series for intermediate magnitudes, and the uniform asymptotic expansions for :math:`I_v(z)` and :math:`J_v(z)` for large orders. Backward recurrence is used to generate sequences or reduce orders when necessary. The calculations above are done in the right half plane and continued into the left half plane by the formula, .. math:: I_v(z \exp(\pm\imath\pi)) = \exp(\pm\pi v) I_v(z) (valid when the real part of `z` is positive). For negative `v`, the formula .. math:: I_{-v}(z) = I_v(z) + \frac{2}{\pi} \sin(\pi v) K_v(z) is used, where :math:`K_v(z)` is the modified Bessel function of the second kind, evaluated using the AMOS routine `zbesk`. See also -------- ive : This function with leading exponential behavior stripped off. i0 : Faster version of this function for order 0. i1 : Faster version of this function for order 1. References ---------- .. [1] Temme, Journal of Computational Physics, vol 21, 343 (1976) .. [2] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions of a Complex Argument and Nonnegative Order", http://netlib.org/amos/ Examples -------- Evaluate the function of order 0 at one point. >>> from scipy.special import iv >>> iv(0, 1.) 1.2660658777520084 Evaluate the function at one point for different orders. >>> iv(0, 1.), iv(1, 1.), iv(1.5, 1.) (1.2660658777520084, 0.565159103992485, 0.2935253263474798) The evaluation for different orders can be carried out in one call by providing a list or NumPy array as argument for the `v` parameter: >>> iv([0, 1, 1.5], 1.) array([1.26606588, 0.5651591 , 0.29352533]) Evaluate the function at several points for order 0 by providing an array for `z`. >>> import numpy as np >>> points = np.array([-2., 0., 3.]) >>> iv(0, points) array([2.2795853 , 1. , 4.88079259]) If `z` is an array, the order parameter `v` must be broadcastable to the correct shape if different orders shall be computed in one call. To calculate the orders 0 and 1 for an 1D array: >>> orders = np.array([[0], [1]]) >>> orders.shape (2, 1) >>> iv(orders, points) array([[ 2.2795853 , 1. , 4.88079259], [-1.59063685, 0. , 3.95337022]]) Plot the functions of order 0 to 3 from -5 to 5. >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots() >>> x = np.linspace(-5., 5., 1000) >>> for i in range(4): ... ax.plot(x, iv(i, x), label=f'$I_{i!r}$') >>> ax.legend() >>> plt.show()iveive(v, z, out=None) Exponentially scaled modified Bessel function of the first kind. Defined as:: ive(v, z) = iv(v, z) * exp(-abs(z.real)) For imaginary numbers without a real part, returns the unscaled Bessel function of the first kind `iv`. Parameters ---------- v : array_like of float Order. z : array_like of float or complex Argument. out : ndarray, optional Optional output array for the function values Returns ------- scalar or ndarray Values of the exponentially scaled modified Bessel function. Notes ----- For positive `v`, the AMOS [1]_ `zbesi` routine is called. It uses a power series for small `z`, the asymptotic expansion for large `abs(z)`, the Miller algorithm normalized by the Wronskian and a Neumann series for intermediate magnitudes, and the uniform asymptotic expansions for :math:`I_v(z)` and :math:`J_v(z)` for large orders. Backward recurrence is used to generate sequences or reduce orders when necessary. The calculations above are done in the right half plane and continued into the left half plane by the formula, .. math:: I_v(z \exp(\pm\imath\pi)) = \exp(\pm\pi v) I_v(z) (valid when the real part of `z` is positive). For negative `v`, the formula .. math:: I_{-v}(z) = I_v(z) + \frac{2}{\pi} \sin(\pi v) K_v(z) is used, where :math:`K_v(z)` is the modified Bessel function of the second kind, evaluated using the AMOS routine `zbesk`. `ive` is useful for large arguments `z`: for these, `iv` easily overflows, while `ive` does not due to the exponential scaling. See also -------- iv: Modified Bessel function of the first kind i0e: Faster implementation of this function for order 0 i1e: Faster implementation of this function for order 1 References ---------- .. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions of a Complex Argument and Nonnegative Order", http://netlib.org/amos/ Examples -------- In the following example `iv` returns infinity whereas `ive` still returns a finite number. >>> from scipy.special import iv, ive >>> import numpy as np >>> import matplotlib.pyplot as plt >>> iv(3, 1000.), ive(3, 1000.) (inf, 0.01256056218254712) Evaluate the function at one point for different orders by providing a list or NumPy array as argument for the `v` parameter: >>> ive([0, 1, 1.5], 1.) array([0.46575961, 0.20791042, 0.10798193]) Evaluate the function at several points for order 0 by providing an array for `z`. >>> points = np.array([-2., 0., 3.]) >>> ive(0, points) array([0.30850832, 1. , 0.24300035]) Evaluate the function at several points for different orders by providing arrays for both `v` for `z`. Both arrays have to be broadcastable to the correct shape. To calculate the orders 0, 1 and 2 for a 1D array of points: >>> ive([[0], [1], [2]], points) array([[ 0.30850832, 1. , 0.24300035], [-0.21526929, 0. , 0.19682671], [ 0.09323903, 0. , 0.11178255]]) Plot the functions of order 0 to 3 from -5 to 5. >>> fig, ax = plt.subplots() >>> x = np.linspace(-5., 5., 1000) >>> for i in range(4): ... ax.plot(x, ive(i, x), label=f'$I_{i!r}(z)\cdot e^{{-|z|}}$') >>> ax.legend() >>> ax.set_xlabel(r"$z$") >>> plt.show()j0j0(x, out=None) Bessel function of the first kind of order 0. Parameters ---------- x : array_like Argument (float). out : ndarray, optional Optional output array for the function values Returns ------- J : scalar or ndarray Value of the Bessel function of the first kind of order 0 at `x`. Notes ----- The domain is divided into the intervals [0, 5] and (5, infinity). In the first interval the following rational approximation is used: .. math:: J_0(x) \approx (w - r_1^2)(w - r_2^2) \frac{P_3(w)}{Q_8(w)}, where :math:`w = x^2` and :math:`r_1`, :math:`r_2` are the zeros of :math:`J_0`, and :math:`P_3` and :math:`Q_8` are polynomials of degrees 3 and 8, respectively. In the second interval, the Hankel asymptotic expansion is employed with two rational functions of degree 6/6 and 7/7. This function is a wrapper for the Cephes [1]_ routine `j0`. It should not be confused with the spherical Bessel functions (see `spherical_jn`). See also -------- jv : Bessel function of real order and complex argument. spherical_jn : spherical Bessel functions. References ---------- .. [1] Cephes Mathematical Functions Library, http://www.netlib.org/cephes/ Examples -------- Calculate the function at one point: >>> from scipy.special import j0 >>> j0(1.) 0.7651976865579665 Calculate the function at several points: >>> import numpy as np >>> j0(np.array([-2., 0., 4.])) array([ 0.22389078, 1. , -0.39714981]) Plot the function from -20 to 20. >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots() >>> x = np.linspace(-20., 20., 1000) >>> y = j0(x) >>> ax.plot(x, y) >>> plt.show()j1j1(x, out=None) Bessel function of the first kind of order 1. Parameters ---------- x : array_like Argument (float). out : ndarray, optional Optional output array for the function values Returns ------- J : scalar or ndarray Value of the Bessel function of the first kind of order 1 at `x`. Notes ----- The domain is divided into the intervals [0, 8] and (8, infinity). In the first interval a 24 term Chebyshev expansion is used. In the second, the asymptotic trigonometric representation is employed using two rational functions of degree 5/5. This function is a wrapper for the Cephes [1]_ routine `j1`. It should not be confused with the spherical Bessel functions (see `spherical_jn`). See also -------- jv: Bessel function of the first kind spherical_jn: spherical Bessel functions. References ---------- .. [1] Cephes Mathematical Functions Library, http://www.netlib.org/cephes/ Examples -------- Calculate the function at one point: >>> from scipy.special import j1 >>> j1(1.) 0.44005058574493355 Calculate the function at several points: >>> import numpy as np >>> j1(np.array([-2., 0., 4.])) array([-0.57672481, 0. , -0.06604333]) Plot the function from -20 to 20. >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots() >>> x = np.linspace(-20., 20., 1000) >>> y = j1(x) >>> ax.plot(x, y) >>> plt.show()jvjv(v, z, out=None) Bessel function of the first kind of real order and complex argument. Parameters ---------- v : array_like Order (float). z : array_like Argument (float or complex). out : ndarray, optional Optional output array for the function values Returns ------- J : scalar or ndarray Value of the Bessel function, :math:`J_v(z)`. See also -------- jve : :math:`J_v` with leading exponential behavior stripped off. spherical_jn : spherical Bessel functions. j0 : faster version of this function for order 0. j1 : faster version of this function for order 1. Notes ----- For positive `v` values, the computation is carried out using the AMOS [1]_ `zbesj` routine, which exploits the connection to the modified Bessel function :math:`I_v`, .. math:: J_v(z) = \exp(v\pi\imath/2) I_v(-\imath z)\qquad (\Im z > 0) J_v(z) = \exp(-v\pi\imath/2) I_v(\imath z)\qquad (\Im z < 0) For negative `v` values the formula, .. math:: J_{-v}(z) = J_v(z) \cos(\pi v) - Y_v(z) \sin(\pi v) is used, where :math:`Y_v(z)` is the Bessel function of the second kind, computed using the AMOS routine `zbesy`. Note that the second term is exactly zero for integer `v`; to improve accuracy the second term is explicitly omitted for `v` values such that `v = floor(v)`. Not to be confused with the spherical Bessel functions (see `spherical_jn`). References ---------- .. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions of a Complex Argument and Nonnegative Order", http://netlib.org/amos/ Examples -------- Evaluate the function of order 0 at one point. >>> from scipy.special import jv >>> jv(0, 1.) 0.7651976865579666 Evaluate the function at one point for different orders. >>> jv(0, 1.), jv(1, 1.), jv(1.5, 1.) (0.7651976865579666, 0.44005058574493355, 0.24029783912342725) The evaluation for different orders can be carried out in one call by providing a list or NumPy array as argument for the `v` parameter: >>> jv([0, 1, 1.5], 1.) array([0.76519769, 0.44005059, 0.24029784]) Evaluate the function at several points for order 0 by providing an array for `z`. >>> import numpy as np >>> points = np.array([-2., 0., 3.]) >>> jv(0, points) array([ 0.22389078, 1. , -0.26005195]) If `z` is an array, the order parameter `v` must be broadcastable to the correct shape if different orders shall be computed in one call. To calculate the orders 0 and 1 for an 1D array: >>> orders = np.array([[0], [1]]) >>> orders.shape (2, 1) >>> jv(orders, points) array([[ 0.22389078, 1. , -0.26005195], [-0.57672481, 0. , 0.33905896]]) Plot the functions of order 0 to 3 from -10 to 10. >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots() >>> x = np.linspace(-10., 10., 1000) >>> for i in range(4): ... ax.plot(x, jv(i, x), label=f'$J_{i!r}$') >>> ax.legend() >>> plt.show()jvejve(v, z, out=None) Exponentially scaled Bessel function of the first kind of order `v`. Defined as:: jve(v, z) = jv(v, z) * exp(-abs(z.imag)) Parameters ---------- v : array_like Order (float). z : array_like Argument (float or complex). out : ndarray, optional Optional output array for the function values Returns ------- J : scalar or ndarray Value of the exponentially scaled Bessel function. See also -------- jv: Unscaled Bessel function of the first kind Notes ----- For positive `v` values, the computation is carried out using the AMOS [1]_ `zbesj` routine, which exploits the connection to the modified Bessel function :math:`I_v`, .. math:: J_v(z) = \exp(v\pi\imath/2) I_v(-\imath z)\qquad (\Im z > 0) J_v(z) = \exp(-v\pi\imath/2) I_v(\imath z)\qquad (\Im z < 0) For negative `v` values the formula, .. math:: J_{-v}(z) = J_v(z) \cos(\pi v) - Y_v(z) \sin(\pi v) is used, where :math:`Y_v(z)` is the Bessel function of the second kind, computed using the AMOS routine `zbesy`. Note that the second term is exactly zero for integer `v`; to improve accuracy the second term is explicitly omitted for `v` values such that `v = floor(v)`. Exponentially scaled Bessel functions are useful for large arguments `z`: for these, the unscaled Bessel functions can easily under-or overflow. References ---------- .. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions of a Complex Argument and Nonnegative Order", http://netlib.org/amos/ Examples -------- Compare the output of `jv` and `jve` for large complex arguments for `z` by computing their values for order ``v=1`` at ``z=1000j``. We see that `jv` overflows but `jve` returns a finite number: >>> import numpy as np >>> from scipy.special import jv, jve >>> v = 1 >>> z = 1000j >>> jv(v, z), jve(v, z) ((inf+infj), (7.721967686709077e-19+0.012610930256928629j)) For real arguments for `z`, `jve` returns the same as `jv`. >>> v, z = 1, 1000 >>> jv(v, z), jve(v, z) (0.004728311907089523, 0.004728311907089523) The function can be evaluated for several orders at the same time by providing a list or NumPy array for `v`: >>> jve([1, 3, 5], 1j) array([1.27304208e-17+2.07910415e-01j, -4.99352086e-19-8.15530777e-03j, 6.11480940e-21+9.98657141e-05j]) In the same way, the function can be evaluated at several points in one call by providing a list or NumPy array for `z`: >>> jve(1, np.array([1j, 2j, 3j])) array([1.27308412e-17+0.20791042j, 1.31814423e-17+0.21526929j, 1.20521602e-17+0.19682671j]) It is also possible to evaluate several orders at several points at the same time by providing arrays for `v` and `z` with compatible shapes for broadcasting. Compute `jve` for two different orders `v` and three points `z` resulting in a 2x3 array. >>> v = np.array([[1], [3]]) >>> z = np.array([1j, 2j, 3j]) >>> v.shape, z.shape ((2, 1), (3,)) >>> jve(v, z) array([[1.27304208e-17+0.20791042j, 1.31810070e-17+0.21526929j, 1.20517622e-17+0.19682671j], [-4.99352086e-19-0.00815531j, -1.76289571e-18-0.02879122j, -2.92578784e-18-0.04778332j]])k0k0(x, out=None) Modified Bessel function of the second kind of order 0, :math:`K_0`. This function is also sometimes referred to as the modified Bessel function of the third kind of order 0. Parameters ---------- x : array_like Argument (float). out : ndarray, optional Optional output array for the function values Returns ------- K : scalar or ndarray Value of the modified Bessel function :math:`K_0` at `x`. Notes ----- The range is partitioned into the two intervals [0, 2] and (2, infinity). Chebyshev polynomial expansions are employed in each interval. This function is a wrapper for the Cephes [1]_ routine `k0`. See also -------- kv: Modified Bessel function of the second kind of any order k0e: Exponentially scaled modified Bessel function of the second kind References ---------- .. [1] Cephes Mathematical Functions Library, http://www.netlib.org/cephes/ Examples -------- Calculate the function at one point: >>> from scipy.special import k0 >>> k0(1.) 0.42102443824070823 Calculate the function at several points: >>> import numpy as np >>> k0(np.array([0.5, 2., 3.])) array([0.92441907, 0.11389387, 0.0347395 ]) Plot the function from 0 to 10. >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots() >>> x = np.linspace(0., 10., 1000) >>> y = k0(x) >>> ax.plot(x, y) >>> plt.show()k0ek0e(x, out=None) Exponentially scaled modified Bessel function K of order 0 Defined as:: k0e(x) = exp(x) * k0(x). Parameters ---------- x : array_like Argument (float) out : ndarray, optional Optional output array for the function values Returns ------- K : scalar or ndarray Value of the exponentially scaled modified Bessel function K of order 0 at `x`. Notes ----- The range is partitioned into the two intervals [0, 2] and (2, infinity). Chebyshev polynomial expansions are employed in each interval. This function is a wrapper for the Cephes [1]_ routine `k0e`. `k0e` is useful for large arguments: for these, `k0` easily underflows. See also -------- kv: Modified Bessel function of the second kind of any order k0: Modified Bessel function of the second kind References ---------- .. [1] Cephes Mathematical Functions Library, http://www.netlib.org/cephes/ Examples -------- In the following example `k0` returns 0 whereas `k0e` still returns a useful finite number: >>> from scipy.special import k0, k0e >>> k0(1000.), k0e(1000) (0., 0.03962832160075422) Calculate the function at several points by providing a NumPy array or list for `x`: >>> import numpy as np >>> k0e(np.array([0.5, 2., 3.])) array([1.52410939, 0.84156822, 0.6977616 ]) Plot the function from 0 to 10. >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots() >>> x = np.linspace(0., 10., 1000) >>> y = k0e(x) >>> ax.plot(x, y) >>> plt.show()k1k1(x, out=None) Modified Bessel function of the second kind of order 1, :math:`K_1(x)`. Parameters ---------- x : array_like Argument (float) out : ndarray, optional Optional output array for the function values Returns ------- K : scalar or ndarray Value of the modified Bessel function K of order 1 at `x`. Notes ----- The range is partitioned into the two intervals [0, 2] and (2, infinity). Chebyshev polynomial expansions are employed in each interval. This function is a wrapper for the Cephes [1]_ routine `k1`. See also -------- kv: Modified Bessel function of the second kind of any order k1e: Exponentially scaled modified Bessel function K of order 1 References ---------- .. [1] Cephes Mathematical Functions Library, http://www.netlib.org/cephes/ Examples -------- Calculate the function at one point: >>> from scipy.special import k1 >>> k1(1.) 0.6019072301972346 Calculate the function at several points: >>> import numpy as np >>> k1(np.array([0.5, 2., 3.])) array([1.65644112, 0.13986588, 0.04015643]) Plot the function from 0 to 10. >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots() >>> x = np.linspace(0., 10., 1000) >>> y = k1(x) >>> ax.plot(x, y) >>> plt.show()k1ek1e(x, out=None) Exponentially scaled modified Bessel function K of order 1 Defined as:: k1e(x) = exp(x) * k1(x) Parameters ---------- x : array_like Argument (float) out : ndarray, optional Optional output array for the function values Returns ------- K : scalar or ndarray Value of the exponentially scaled modified Bessel function K of order 1 at `x`. Notes ----- The range is partitioned into the two intervals [0, 2] and (2, infinity). Chebyshev polynomial expansions are employed in each interval. This function is a wrapper for the Cephes [1]_ routine `k1e`. See also -------- kv: Modified Bessel function of the second kind of any order k1: Modified Bessel function of the second kind of order 1 References ---------- .. [1] Cephes Mathematical Functions Library, http://www.netlib.org/cephes/ Examples -------- In the following example `k1` returns 0 whereas `k1e` still returns a useful floating point number. >>> from scipy.special import k1, k1e >>> k1(1000.), k1e(1000.) (0., 0.03964813081296021) Calculate the function at several points by providing a NumPy array or list for `x`: >>> import numpy as np >>> k1e(np.array([0.5, 2., 3.])) array([2.73100971, 1.03347685, 0.80656348]) Plot the function from 0 to 10. >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots() >>> x = np.linspace(0., 10., 1000) >>> y = k1e(x) >>> ax.plot(x, y) >>> plt.show()keikei(x, out=None) Kelvin function kei. Defined as .. math:: \mathrm{kei}(x) = \Im[K_0(x e^{\pi i / 4})] where :math:`K_0` is the modified Bessel function of the second kind (see `kv`). See [dlmf]_ for more details. Parameters ---------- x : array_like Real argument. out : ndarray, optional Optional output array for the function results. Returns ------- scalar or ndarray Values of the Kelvin function. See Also -------- ker : the corresponding real part keip : the derivative of kei kv : modified Bessel function of the second kind References ---------- .. [dlmf] NIST, Digital Library of Mathematical Functions, https://dlmf.nist.gov/10.61 Examples -------- It can be expressed using the modified Bessel function of the second kind. >>> import numpy as np >>> import scipy.special as sc >>> x = np.array([1.0, 2.0, 3.0, 4.0]) >>> sc.kv(0, x * np.exp(np.pi * 1j / 4)).imag array([-0.49499464, -0.20240007, -0.05112188, 0.0021984 ]) >>> sc.kei(x) array([-0.49499464, -0.20240007, -0.05112188, 0.0021984 ])keipkeip(x, out=None) Derivative of the Kelvin function kei. Parameters ---------- x : array_like Real argument. out : ndarray, optional Optional output array for the function results. Returns ------- scalar or ndarray The values of the derivative of kei. See Also -------- kei References ---------- .. [dlmf] NIST, Digital Library of Mathematical Functions, https://dlmf.nist.gov/10#PT5kelvinkelvin(x, out=None) Kelvin functions as complex numbers Parameters ---------- x : array_like Argument out : tuple of ndarray, optional Optional output arrays for the function values Returns ------- Be, Ke, Bep, Kep : 4-tuple of scalar or ndarray The tuple (Be, Ke, Bep, Kep) contains complex numbers representing the real and imaginary Kelvin functions and their derivatives evaluated at `x`. For example, kelvin(x)[0].real = ber x and kelvin(x)[0].imag = bei x with similar relationships for ker and kei.kerker(x, out=None) Kelvin function ker. Defined as .. math:: \mathrm{ker}(x) = \Re[K_0(x e^{\pi i / 4})] Where :math:`K_0` is the modified Bessel function of the second kind (see `kv`). See [dlmf]_ for more details. Parameters ---------- x : array_like Real argument. out : ndarray, optional Optional output array for the function results. Returns ------- scalar or ndarray Values of the Kelvin function. See Also -------- kei : the corresponding imaginary part kerp : the derivative of ker kv : modified Bessel function of the second kind References ---------- .. [dlmf] NIST, Digital Library of Mathematical Functions, https://dlmf.nist.gov/10.61 Examples -------- It can be expressed using the modified Bessel function of the second kind. >>> import numpy as np >>> import scipy.special as sc >>> x = np.array([1.0, 2.0, 3.0, 4.0]) >>> sc.kv(0, x * np.exp(np.pi * 1j / 4)).real array([ 0.28670621, -0.04166451, -0.06702923, -0.03617885]) >>> sc.ker(x) array([ 0.28670621, -0.04166451, -0.06702923, -0.03617885])kerpkerp(x, out=None) Derivative of the Kelvin function ker. Parameters ---------- x : array_like Real argument. out : ndarray, optional Optional output array for the function results. Returns ------- scalar or ndarray Values of the derivative of ker. See Also -------- ker References ---------- .. [dlmf] NIST, Digital Library of Mathematical Functions, https://dlmf.nist.gov/10#PT5kl_divkl_div(x, y, out=None) Elementwise function for computing Kullback-Leibler divergence. .. math:: \mathrm{kl\_div}(x, y) = \begin{cases} x \log(x / y) - x + y & x > 0, y > 0 \\ y & x = 0, y \ge 0 \\ \infty & \text{otherwise} \end{cases} Parameters ---------- x, y : array_like Real arguments out : ndarray, optional Optional output array for the function results Returns ------- scalar or ndarray Values of the Kullback-Liebler divergence. See Also -------- entr, rel_entr, scipy.stats.entropy Notes ----- .. versionadded:: 0.15.0 This function is non-negative and is jointly convex in `x` and `y`. The origin of this function is in convex programming; see [1]_ for details. This is why the function contains the extra :math:`-x + y` terms over what might be expected from the Kullback-Leibler divergence. For a version of the function without the extra terms, see `rel_entr`. References ---------- .. [1] Boyd, Stephen and Lieven Vandenberghe. *Convex optimization*. Cambridge University Press, 2004. :doi:`https://doi.org/10.1017/CBO9780511804441`knkn(n, x, out=None) Modified Bessel function of the second kind of integer order `n` Returns the modified Bessel function of the second kind for integer order `n` at real `z`. These are also sometimes called functions of the third kind, Basset functions, or Macdonald functions. Parameters ---------- n : array_like of int Order of Bessel functions (floats will truncate with a warning) x : array_like of float Argument at which to evaluate the Bessel functions out : ndarray, optional Optional output array for the function results. Returns ------- scalar or ndarray Value of the Modified Bessel function of the second kind, :math:`K_n(x)`. Notes ----- Wrapper for AMOS [1]_ routine `zbesk`. For a discussion of the algorithm used, see [2]_ and the references therein. See Also -------- kv : Same function, but accepts real order and complex argument kvp : Derivative of this function References ---------- .. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions of a Complex Argument and Nonnegative Order", http://netlib.org/amos/ .. [2] Donald E. Amos, "Algorithm 644: A portable package for Bessel functions of a complex argument and nonnegative order", ACM TOMS Vol. 12 Issue 3, Sept. 1986, p. 265 Examples -------- Plot the function of several orders for real input: >>> import numpy as np >>> from scipy.special import kn >>> import matplotlib.pyplot as plt >>> x = np.linspace(0, 5, 1000) >>> for N in range(6): ... plt.plot(x, kn(N, x), label='$K_{}(x)$'.format(N)) >>> plt.ylim(0, 10) >>> plt.legend() >>> plt.title(r'Modified Bessel function of the second kind $K_n(x)$') >>> plt.show() Calculate for a single value at multiple orders: >>> kn([4, 5, 6], 1) array([ 44.23241585, 360.9605896 , 3653.83831186])kolmogikolmogi(p, out=None) Inverse Survival Function of Kolmogorov distribution It is the inverse function to `kolmogorov`. Returns y such that ``kolmogorov(y) == p``. Parameters ---------- p : float array_like Probability out : ndarray, optional Optional output array for the function results Returns ------- scalar or ndarray The value(s) of kolmogi(p) Notes ----- `kolmogorov` is used by `stats.kstest` in the application of the Kolmogorov-Smirnov Goodness of Fit test. For historial reasons this function is exposed in `scpy.special`, but the recommended way to achieve the most accurate CDF/SF/PDF/PPF/ISF computations is to use the `stats.kstwobign` distribution. See Also -------- kolmogorov : The Survival Function for the distribution scipy.stats.kstwobign : Provides the functionality as a continuous distribution smirnov, smirnovi : Functions for the one-sided distribution Examples -------- >>> from scipy.special import kolmogi >>> kolmogi([0, 0.1, 0.25, 0.5, 0.75, 0.9, 1.0]) array([ inf, 1.22384787, 1.01918472, 0.82757356, 0.67644769, 0.57117327, 0. ])kolmogorovkolmogorov(y, out=None) Complementary cumulative distribution (Survival Function) function of Kolmogorov distribution. Returns the complementary cumulative distribution function of Kolmogorov's limiting distribution (``D_n*\sqrt(n)`` as n goes to infinity) of a two-sided test for equality between an empirical and a theoretical distribution. It is equal to the (limit as n->infinity of the) probability that ``sqrt(n) * max absolute deviation > y``. Parameters ---------- y : float array_like Absolute deviation between the Empirical CDF (ECDF) and the target CDF, multiplied by sqrt(n). out : ndarray, optional Optional output array for the function results Returns ------- scalar or ndarray The value(s) of kolmogorov(y) Notes ----- `kolmogorov` is used by `stats.kstest` in the application of the Kolmogorov-Smirnov Goodness of Fit test. For historial reasons this function is exposed in `scpy.special`, but the recommended way to achieve the most accurate CDF/SF/PDF/PPF/ISF computations is to use the `stats.kstwobign` distribution. See Also -------- kolmogi : The Inverse Survival Function for the distribution scipy.stats.kstwobign : Provides the functionality as a continuous distribution smirnov, smirnovi : Functions for the one-sided distribution Examples -------- Show the probability of a gap at least as big as 0, 0.5 and 1.0. >>> import numpy as np >>> from scipy.special import kolmogorov >>> from scipy.stats import kstwobign >>> kolmogorov([0, 0.5, 1.0]) array([ 1. , 0.96394524, 0.26999967]) Compare a sample of size 1000 drawn from a Laplace(0, 1) distribution against the target distribution, a Normal(0, 1) distribution. >>> from scipy.stats import norm, laplace >>> rng = np.random.default_rng() >>> n = 1000 >>> lap01 = laplace(0, 1) >>> x = np.sort(lap01.rvs(n, random_state=rng)) >>> np.mean(x), np.std(x) (-0.05841730131499543, 1.3968109101997568) Construct the Empirical CDF and the K-S statistic Dn. >>> target = norm(0,1) # Normal mean 0, stddev 1 >>> cdfs = target.cdf(x) >>> ecdfs = np.arange(n+1, dtype=float)/n >>> gaps = np.column_stack([cdfs - ecdfs[:n], ecdfs[1:] - cdfs]) >>> Dn = np.max(gaps) >>> Kn = np.sqrt(n) * Dn >>> print('Dn=%f, sqrt(n)*Dn=%f' % (Dn, Kn)) Dn=0.043363, sqrt(n)*Dn=1.371265 >>> print(chr(10).join(['For a sample of size n drawn from a N(0, 1) distribution:', ... ' the approximate Kolmogorov probability that sqrt(n)*Dn>=%f is %f' % (Kn, kolmogorov(Kn)), ... ' the approximate Kolmogorov probability that sqrt(n)*Dn<=%f is %f' % (Kn, kstwobign.cdf(Kn))])) For a sample of size n drawn from a N(0, 1) distribution: the approximate Kolmogorov probability that sqrt(n)*Dn>=1.371265 is 0.046533 the approximate Kolmogorov probability that sqrt(n)*Dn<=1.371265 is 0.953467 Plot the Empirical CDF against the target N(0, 1) CDF. >>> import matplotlib.pyplot as plt >>> plt.step(np.concatenate([[-3], x]), ecdfs, where='post', label='Empirical CDF') >>> x3 = np.linspace(-3, 3, 100) >>> plt.plot(x3, target.cdf(x3), label='CDF for N(0, 1)') >>> plt.ylim([0, 1]); plt.grid(True); plt.legend(); >>> # Add vertical lines marking Dn+ and Dn- >>> iminus, iplus = np.argmax(gaps, axis=0) >>> plt.vlines([x[iminus]], ecdfs[iminus], cdfs[iminus], color='r', linestyle='dashed', lw=4) >>> plt.vlines([x[iplus]], cdfs[iplus], ecdfs[iplus+1], color='r', linestyle='dashed', lw=4) >>> plt.show()kvkv(v, z, out=None) Modified Bessel function of the second kind of real order `v` Returns the modified Bessel function of the second kind for real order `v` at complex `z`. These are also sometimes called functions of the third kind, Basset functions, or Macdonald functions. They are defined as those solutions of the modified Bessel equation for which, .. math:: K_v(x) \sim \sqrt{\pi/(2x)} \exp(-x) as :math:`x \to \infty` [3]_. Parameters ---------- v : array_like of float Order of Bessel functions z : array_like of complex Argument at which to evaluate the Bessel functions out : ndarray, optional Optional output array for the function results Returns ------- scalar or ndarray The results. Note that input must be of complex type to get complex output, e.g. ``kv(3, -2+0j)`` instead of ``kv(3, -2)``. Notes ----- Wrapper for AMOS [1]_ routine `zbesk`. For a discussion of the algorithm used, see [2]_ and the references therein. See Also -------- kve : This function with leading exponential behavior stripped off. kvp : Derivative of this function References ---------- .. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions of a Complex Argument and Nonnegative Order", http://netlib.org/amos/ .. [2] Donald E. Amos, "Algorithm 644: A portable package for Bessel functions of a complex argument and nonnegative order", ACM TOMS Vol. 12 Issue 3, Sept. 1986, p. 265 .. [3] NIST Digital Library of Mathematical Functions, Eq. 10.25.E3. https://dlmf.nist.gov/10.25.E3 Examples -------- Plot the function of several orders for real input: >>> import numpy as np >>> from scipy.special import kv >>> import matplotlib.pyplot as plt >>> x = np.linspace(0, 5, 1000) >>> for N in np.linspace(0, 6, 5): ... plt.plot(x, kv(N, x), label='$K_{{{}}}(x)$'.format(N)) >>> plt.ylim(0, 10) >>> plt.legend() >>> plt.title(r'Modified Bessel function of the second kind $K_\nu(x)$') >>> plt.show() Calculate for a single value at multiple orders: >>> kv([4, 4.5, 5], 1+2j) array([ 0.1992+2.3892j, 2.3493+3.6j , 7.2827+3.8104j])kvekve(v, z, out=None) Exponentially scaled modified Bessel function of the second kind. Returns the exponentially scaled, modified Bessel function of the second kind (sometimes called the third kind) for real order `v` at complex `z`:: kve(v, z) = kv(v, z) * exp(z) Parameters ---------- v : array_like of float Order of Bessel functions z : array_like of complex Argument at which to evaluate the Bessel functions out : ndarray, optional Optional output array for the function results Returns ------- scalar or ndarray The exponentially scaled modified Bessel function of the second kind. Notes ----- Wrapper for AMOS [1]_ routine `zbesk`. For a discussion of the algorithm used, see [2]_ and the references therein. See Also -------- kv : This function without exponential scaling. k0e : Faster version of this function for order 0. k1e : Faster version of this function for order 1. References ---------- .. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions of a Complex Argument and Nonnegative Order", http://netlib.org/amos/ .. [2] Donald E. Amos, "Algorithm 644: A portable package for Bessel functions of a complex argument and nonnegative order", ACM TOMS Vol. 12 Issue 3, Sept. 1986, p. 265 Examples -------- In the following example `kv` returns 0 whereas `kve` still returns a useful finite number. >>> import numpy as np >>> from scipy.special import kv, kve >>> import matplotlib.pyplot as plt >>> kv(3, 1000.), kve(3, 1000.) (0.0, 0.03980696128440973) Evaluate the function at one point for different orders by providing a list or NumPy array as argument for the `v` parameter: >>> kve([0, 1, 1.5], 1.) array([1.14446308, 1.63615349, 2.50662827]) Evaluate the function at several points for order 0 by providing an array for `z`. >>> points = np.array([1., 3., 10.]) >>> kve(0, points) array([1.14446308, 0.6977616 , 0.39163193]) Evaluate the function at several points for different orders by providing arrays for both `v` for `z`. Both arrays have to be broadcastable to the correct shape. To calculate the orders 0, 1 and 2 for a 1D array of points: >>> kve([[0], [1], [2]], points) array([[1.14446308, 0.6977616 , 0.39163193], [1.63615349, 0.80656348, 0.41076657], [4.41677005, 1.23547058, 0.47378525]]) Plot the functions of order 0 to 3 from 0 to 5. >>> fig, ax = plt.subplots() >>> x = np.linspace(0., 5., 1000) >>> for i in range(4): ... ax.plot(x, kve(i, x), label=f'$K_{i!r}(z)\cdot e^z$') >>> ax.legend() >>> ax.set_xlabel(r"$z$") >>> ax.set_ylim(0, 4) >>> ax.set_xlim(0, 5) >>> plt.show()log1plog1p(x, out=None) Calculates log(1 + x) for use when `x` is near zero. Parameters ---------- x : array_like Real or complex valued input. out : ndarray, optional Optional output array for the function results. Returns ------- scalar or ndarray Values of ``log(1 + x)``. See Also -------- expm1, cosm1 Examples -------- >>> import numpy as np >>> import scipy.special as sc It is more accurate than using ``log(1 + x)`` directly for ``x`` near 0. Note that in the below example ``1 + 1e-17 == 1`` to double precision. >>> sc.log1p(1e-17) 1e-17 >>> np.log(1 + 1e-17) 0.0log_expit(x, out=None) Logarithm of the logistic sigmoid function. The SciPy implementation of the logistic sigmoid function is `scipy.special.expit`, so this function is called ``log_expit``. The function is mathematically equivalent to ``log(expit(x))``, but is formulated to avoid loss of precision for inputs with large (positive or negative) magnitude. Parameters ---------- x : array_like The values to apply ``log_expit`` to element-wise. out : ndarray, optional Optional output array for the function results Returns ------- out : scalar or ndarray The computed values, an ndarray of the same shape as ``x``. See Also -------- expit Notes ----- As a ufunc, ``log_expit`` takes a number of optional keyword arguments. For more information see `ufuncs `_ .. versionadded:: 1.8.0 Examples -------- >>> import numpy as np >>> from scipy.special import log_expit, expit >>> log_expit([-3.0, 0.25, 2.5, 5.0]) array([-3.04858735, -0.57593942, -0.07888973, -0.00671535]) Large negative values: >>> log_expit([-100, -500, -1000]) array([ -100., -500., -1000.]) Note that ``expit(-1000)`` returns 0, so the naive implementation ``log(expit(-1000))`` return ``-inf``. Large positive values: >>> log_expit([29, 120, 400]) array([-2.54366565e-013, -7.66764807e-053, -1.91516960e-174]) Compare that to the naive implementation: >>> np.log(expit([29, 120, 400])) array([-2.54463117e-13, 0.00000000e+00, 0.00000000e+00]) The first value is accurate to only 3 digits, and the larger inputs lose all precision and return 0.log_ndtrlog_ndtr(x, out=None) Logarithm of Gaussian cumulative distribution function. Returns the log of the area under the standard Gaussian probability density function, integrated from minus infinity to `x`:: log(1/sqrt(2*pi) * integral(exp(-t**2 / 2), t=-inf..x)) Parameters ---------- x : array_like, real or complex Argument out : ndarray, optional Optional output array for the function results Returns ------- scalar or ndarray The value of the log of the normal CDF evaluated at `x` See Also -------- erf erfc scipy.stats.norm ndtr Examples -------- >>> import numpy as np >>> from scipy.special import log_ndtr, ndtr The benefit of ``log_ndtr(x)`` over the naive implementation ``np.log(ndtr(x))`` is most evident with moderate to large positive values of ``x``: >>> x = np.array([6, 7, 9, 12, 15, 25]) >>> log_ndtr(x) array([-9.86587646e-010, -1.27981254e-012, -1.12858841e-019, -1.77648211e-033, -3.67096620e-051, -3.05669671e-138]) The results of the naive calculation for the moderate ``x`` values have only 5 or 6 correct significant digits. For values of ``x`` greater than approximately 8.3, the naive expression returns 0: >>> np.log(ndtr(x)) array([-9.86587701e-10, -1.27986510e-12, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00])loggamma(z, out=None) Principal branch of the logarithm of the gamma function. Defined to be :math:`\log(\Gamma(x))` for :math:`x > 0` and extended to the complex plane by analytic continuation. The function has a single branch cut on the negative real axis. .. versionadded:: 0.18.0 Parameters ---------- z : array_like Values in the complex plane at which to compute ``loggamma`` out : ndarray, optional Output array for computed values of ``loggamma`` Returns ------- loggamma : scalar or ndarray Values of ``loggamma`` at z. Notes ----- It is not generally true that :math:`\log\Gamma(z) = \log(\Gamma(z))`, though the real parts of the functions do agree. The benefit of not defining `loggamma` as :math:`\log(\Gamma(z))` is that the latter function has a complicated branch cut structure whereas `loggamma` is analytic except for on the negative real axis. The identities .. math:: \exp(\log\Gamma(z)) &= \Gamma(z) \\ \log\Gamma(z + 1) &= \log(z) + \log\Gamma(z) make `loggamma` useful for working in complex logspace. On the real line `loggamma` is related to `gammaln` via ``exp(loggamma(x + 0j)) = gammasgn(x)*exp(gammaln(x))``, up to rounding error. The implementation here is based on [hare1997]_. See also -------- gammaln : logarithm of the absolute value of the gamma function gammasgn : sign of the gamma function References ---------- .. [hare1997] D.E.G. Hare, *Computing the Principal Branch of log-Gamma*, Journal of Algorithms, Volume 25, Issue 2, November 1997, pages 221-236.logit(x, out=None) Logit ufunc for ndarrays. The logit function is defined as logit(p) = log(p/(1-p)). Note that logit(0) = -inf, logit(1) = inf, and logit(p) for p<0 or p>1 yields nan. Parameters ---------- x : ndarray The ndarray to apply logit to element-wise. out : ndarray, optional Optional output array for the function results Returns ------- scalar or ndarray An ndarray of the same shape as x. Its entries are logit of the corresponding entry of x. See Also -------- expit Notes ----- As a ufunc logit takes a number of optional keyword arguments. For more information see `ufuncs `_ .. versionadded:: 0.10.0 Examples -------- >>> import numpy as np >>> from scipy.special import logit, expit >>> logit([0, 0.25, 0.5, 0.75, 1]) array([ -inf, -1.09861229, 0. , 1.09861229, inf]) `expit` is the inverse of `logit`: >>> expit(logit([0.1, 0.75, 0.999])) array([ 0.1 , 0.75 , 0.999]) Plot logit(x) for x in [0, 1]: >>> import matplotlib.pyplot as plt >>> x = np.linspace(0, 1, 501) >>> y = logit(x) >>> plt.plot(x, y) >>> plt.grid() >>> plt.ylim(-6, 6) >>> plt.xlabel('x') >>> plt.title('logit(x)') >>> plt.show()lpmvlpmv(m, v, x, out=None) Associated Legendre function of integer order and real degree. Defined as .. math:: P_v^m = (-1)^m (1 - x^2)^{m/2} \frac{d^m}{dx^m} P_v(x) where .. math:: P_v = \sum_{k = 0}^\infty \frac{(-v)_k (v + 1)_k}{(k!)^2} \left(\frac{1 - x}{2}\right)^k is the Legendre function of the first kind. Here :math:`(\cdot)_k` is the Pochhammer symbol; see `poch`. Parameters ---------- m : array_like Order (int or float). If passed a float not equal to an integer the function returns NaN. v : array_like Degree (float). x : array_like Argument (float). Must have ``|x| <= 1``. out : ndarray, optional Optional output array for the function results Returns ------- pmv : scalar or ndarray Value of the associated Legendre function. See Also -------- lpmn : Compute the associated Legendre function for all orders ``0, ..., m`` and degrees ``0, ..., n``. clpmn : Compute the associated Legendre function at complex arguments. Notes ----- Note that this implementation includes the Condon-Shortley phase. References ---------- .. [1] Zhang, Jin, "Computation of Special Functions", John Wiley and Sons, Inc, 1996.mathieu_amathieu_a(m, q, out=None) Characteristic value of even Mathieu functions Parameters ---------- m : array_like Order of the function q : array_like Parameter of the function out : ndarray, optional Optional output array for the function results Returns ------- scalar or ndarray Characteristic value for the even solution, ``ce_m(z, q)``, of Mathieu's equation. See Also -------- mathieu_b, mathieu_cem, mathieu_semmathieu_bmathieu_b(m, q, out=None) Characteristic value of odd Mathieu functions Parameters ---------- m : array_like Order of the function q : array_like Parameter of the function out : ndarray, optional Optional output array for the function results Returns ------- scalar or ndarray Characteristic value for the odd solution, ``se_m(z, q)``, of Mathieu's equation. See Also -------- mathieu_a, mathieu_cem, mathieu_semmathieu_cemmathieu_cem(m, q, x, out=None) Even Mathieu function and its derivative Returns the even Mathieu function, ``ce_m(x, q)``, of order `m` and parameter `q` evaluated at `x` (given in degrees). Also returns the derivative with respect to `x` of ce_m(x, q) Parameters ---------- m : array_like Order of the function q : array_like Parameter of the function x : array_like Argument of the function, *given in degrees, not radians* out : tuple of ndarray, optional Optional output arrays for the function results Returns ------- y : scalar or ndarray Value of the function yp : scalar or ndarray Value of the derivative vs x See Also -------- mathieu_a, mathieu_b, mathieu_semmathieu_modcem1mathieu_modcem1(m, q, x, out=None) Even modified Mathieu function of the first kind and its derivative Evaluates the even modified Mathieu function of the first kind, ``Mc1m(x, q)``, and its derivative at `x` for order `m` and parameter `q`. Parameters ---------- m : array_like Order of the function q : array_like Parameter of the function x : array_like Argument of the function, *given in degrees, not radians* out : tuple of ndarray, optional Optional output arrays for the function results Returns ------- y : scalar or ndarray Value of the function yp : scalar or ndarray Value of the derivative vs x See Also -------- mathieu_modsem1mathieu_modcem2mathieu_modcem2(m, q, x, out=None) Even modified Mathieu function of the second kind and its derivative Evaluates the even modified Mathieu function of the second kind, Mc2m(x, q), and its derivative at `x` (given in degrees) for order `m` and parameter `q`. Parameters ---------- m : array_like Order of the function q : array_like Parameter of the function x : array_like Argument of the function, *given in degrees, not radians* out : tuple of ndarray, optional Optional output arrays for the function results Returns ------- y : scalar or ndarray Value of the function yp : scalar or ndarray Value of the derivative vs x See Also -------- mathieu_modsem2mathieu_modsem1mathieu_modsem1(m, q, x, out=None) Odd modified Mathieu function of the first kind and its derivative Evaluates the odd modified Mathieu function of the first kind, Ms1m(x, q), and its derivative at `x` (given in degrees) for order `m` and parameter `q`. Parameters ---------- m : array_like Order of the function q : array_like Parameter of the function x : array_like Argument of the function, *given in degrees, not radians* out : tuple of ndarray, optional Optional output arrays for the function results Returns ------- y : scalar or ndarray Value of the function yp : scalar or ndarray Value of the derivative vs x See Also -------- mathieu_modcem1mathieu_modsem2(m, q, x, out=None) Odd modified Mathieu function of the second kind and its derivative Evaluates the odd modified Mathieu function of the second kind, Ms2m(x, q), and its derivative at `x` (given in degrees) for order `m` and parameter q. Parameters ---------- m : array_like Order of the function q : array_like Parameter of the function x : array_like Argument of the function, *given in degrees, not radians* out : tuple of ndarray, optional Optional output arrays for the function results Returns ------- y : scalar or ndarray Value of the function yp : scalar or ndarray Value of the derivative vs x See Also -------- mathieu_modcem2mathieu_semmathieu_sem(m, q, x, out=None) Odd Mathieu function and its derivative Returns the odd Mathieu function, se_m(x, q), of order `m` and parameter `q` evaluated at `x` (given in degrees). Also returns the derivative with respect to `x` of se_m(x, q). Parameters ---------- m : array_like Order of the function q : array_like Parameter of the function x : array_like Argument of the function, *given in degrees, not radians*. out : tuple of ndarray, optional Optional output arrays for the function results Returns ------- y : scalar or ndarray Value of the function yp : scalar or ndarray Value of the derivative vs x See Also -------- mathieu_a, mathieu_b, mathieu_cemmodfresnelmmodfresnelm(x, out=None) Modified Fresnel negative integrals Parameters ---------- x : array_like Function argument out : tuple of ndarray, optional Optional output arrays for the function results Returns ------- fm : scalar or ndarray Integral ``F_-(x)``: ``integral(exp(-1j*t*t), t=x..inf)`` km : scalar or ndarray Integral ``K_-(x)``: ``1/sqrt(pi)*exp(1j*(x*x+pi/4))*fp`` See Also -------- modfresnelpmodfresnelpmodfresnelp(x, out=None) Modified Fresnel positive integrals Parameters ---------- x : array_like Function argument out : tuple of ndarray, optional Optional output arrays for the function results Returns ------- fp : scalar or ndarray Integral ``F_+(x)``: ``integral(exp(1j*t*t), t=x..inf)`` kp : scalar or ndarray Integral ``K_+(x)``: ``1/sqrt(pi)*exp(-1j*(x*x+pi/4))*fp`` See Also -------- modfresnelmmodstruvemodstruve(v, x, out=None) Modified Struve function. Return the value of the modified Struve function of order `v` at `x`. The modified Struve function is defined as, .. math:: L_v(x) = -\imath \exp(-\pi\imath v/2) H_v(\imath x), where :math:`H_v` is the Struve function. Parameters ---------- v : array_like Order of the modified Struve function (float). x : array_like Argument of the Struve function (float; must be positive unless `v` is an integer). out : ndarray, optional Optional output array for the function results Returns ------- L : scalar or ndarray Value of the modified Struve function of order `v` at `x`. Notes ----- Three methods discussed in [1]_ are used to evaluate the function: - power series - expansion in Bessel functions (if :math:`|x| < |v| + 20`) - asymptotic large-x expansion (if :math:`x \geq 0.7v + 12`) Rounding errors are estimated based on the largest terms in the sums, and the result associated with the smallest error is returned. See also -------- struve References ---------- .. [1] NIST Digital Library of Mathematical Functions https://dlmf.nist.gov/11 Examples -------- Calculate the modified Struve function of order 1 at 2. >>> import numpy as np >>> from scipy.special import modstruve >>> import matplotlib.pyplot as plt >>> modstruve(1, 2.) 1.102759787367716 Calculate the modified Struve function at 2 for orders 1, 2 and 3 by providing a list for the order parameter `v`. >>> modstruve([1, 2, 3], 2.) array([1.10275979, 0.41026079, 0.11247294]) Calculate the modified Struve function of order 1 for several points by providing an array for `x`. >>> points = np.array([2., 5., 8.]) >>> modstruve(1, points) array([ 1.10275979, 23.72821578, 399.24709139]) Compute the modified Struve function for several orders at several points by providing arrays for `v` and `z`. The arrays have to be broadcastable to the correct shapes. >>> orders = np.array([[1], [2], [3]]) >>> points.shape, orders.shape ((3,), (3, 1)) >>> modstruve(orders, points) array([[1.10275979e+00, 2.37282158e+01, 3.99247091e+02], [4.10260789e-01, 1.65535979e+01, 3.25973609e+02], [1.12472937e-01, 9.42430454e+00, 2.33544042e+02]]) Plot the modified Struve functions of order 0 to 3 from -5 to 5. >>> fig, ax = plt.subplots() >>> x = np.linspace(-5., 5., 1000) >>> for i in range(4): ... ax.plot(x, modstruve(i, x), label=f'$L_{i!r}$') >>> ax.legend(ncol=2) >>> ax.set_xlim(-5, 5) >>> ax.set_title(r"Modified Struve functions $L_{\nu}$") >>> plt.show()nbdtrnbdtr(k, n, p, out=None) Negative binomial cumulative distribution function. Returns the sum of the terms 0 through `k` of the negative binomial distribution probability mass function, .. math:: F = \sum_{j=0}^k {{n + j - 1}\choose{j}} p^n (1 - p)^j. In a sequence of Bernoulli trials with individual success probabilities `p`, this is the probability that `k` or fewer failures precede the nth success. Parameters ---------- k : array_like The maximum number of allowed failures (nonnegative int). n : array_like The target number of successes (positive int). p : array_like Probability of success in a single event (float). out : ndarray, optional Optional output array for the function results Returns ------- F : scalar or ndarray The probability of `k` or fewer failures before `n` successes in a sequence of events with individual success probability `p`. See also -------- nbdtrc : Negative binomial survival function nbdtrik : Negative binomial quantile function scipy.stats.nbinom : Negative binomial distribution Notes ----- If floating point values are passed for `k` or `n`, they will be truncated to integers. The terms are not summed directly; instead the regularized incomplete beta function is employed, according to the formula, .. math:: \mathrm{nbdtr}(k, n, p) = I_{p}(n, k + 1). Wrapper for the Cephes [1]_ routine `nbdtr`. The negative binomial distribution is also available as `scipy.stats.nbinom`. Using `nbdtr` directly can improve performance compared to the ``cdf`` method of `scipy.stats.nbinom` (see last example). References ---------- .. [1] Cephes Mathematical Functions Library, http://www.netlib.org/cephes/ Examples -------- Compute the function for ``k=10`` and ``n=5`` at ``p=0.5``. >>> import numpy as np >>> from scipy.special import nbdtr >>> nbdtr(10, 5, 0.5) 0.940765380859375 Compute the function for ``n=10`` and ``p=0.5`` at several points by providing a NumPy array or list for `k`. >>> nbdtr([5, 10, 15], 10, 0.5) array([0.15087891, 0.58809853, 0.88523853]) Plot the function for four different parameter sets. >>> import matplotlib.pyplot as plt >>> k = np.arange(130) >>> n_parameters = [20, 20, 20, 80] >>> p_parameters = [0.2, 0.5, 0.8, 0.5] >>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot'] >>> parameters_list = list(zip(p_parameters, n_parameters, ... linestyles)) >>> fig, ax = plt.subplots(figsize=(8, 8)) >>> for parameter_set in parameters_list: ... p, n, style = parameter_set ... nbdtr_vals = nbdtr(k, n, p) ... ax.plot(k, nbdtr_vals, label=rf"$n={n},\, p={p}$", ... ls=style) >>> ax.legend() >>> ax.set_xlabel("$k$") >>> ax.set_title("Negative binomial cumulative distribution function") >>> plt.show() The negative binomial distribution is also available as `scipy.stats.nbinom`. Using `nbdtr` directly can be much faster than calling the ``cdf`` method of `scipy.stats.nbinom`, especially for small arrays or individual values. To get the same results one must use the following parametrization: ``nbinom(n, p).cdf(k)=nbdtr(k, n, p)``. >>> from scipy.stats import nbinom >>> k, n, p = 5, 3, 0.5 >>> nbdtr_res = nbdtr(k, n, p) # this will often be faster than below >>> stats_res = nbinom(n, p).cdf(k) >>> stats_res, nbdtr_res # test that results are equal (0.85546875, 0.85546875) `nbdtr` can evaluate different parameter sets by providing arrays with shapes compatible for broadcasting for `k`, `n` and `p`. Here we compute the function for three different `k` at four locations `p`, resulting in a 3x4 array. >>> k = np.array([[5], [10], [15]]) >>> p = np.array([0.3, 0.5, 0.7, 0.9]) >>> k.shape, p.shape ((3, 1), (4,)) >>> nbdtr(k, 5, p) array([[0.15026833, 0.62304687, 0.95265101, 0.9998531 ], [0.48450894, 0.94076538, 0.99932777, 0.99999999], [0.76249222, 0.99409103, 0.99999445, 1. ]])nbdtrcnbdtrc(k, n, p, out=None) Negative binomial survival function. Returns the sum of the terms `k + 1` to infinity of the negative binomial distribution probability mass function, .. math:: F = \sum_{j=k + 1}^\infty {{n + j - 1}\choose{j}} p^n (1 - p)^j. In a sequence of Bernoulli trials with individual success probabilities `p`, this is the probability that more than `k` failures precede the nth success. Parameters ---------- k : array_like The maximum number of allowed failures (nonnegative int). n : array_like The target number of successes (positive int). p : array_like Probability of success in a single event (float). out : ndarray, optional Optional output array for the function results Returns ------- F : scalar or ndarray The probability of `k + 1` or more failures before `n` successes in a sequence of events with individual success probability `p`. See also -------- nbdtr : Negative binomial cumulative distribution function nbdtrik : Negative binomial percentile function scipy.stats.nbinom : Negative binomial distribution Notes ----- If floating point values are passed for `k` or `n`, they will be truncated to integers. The terms are not summed directly; instead the regularized incomplete beta function is employed, according to the formula, .. math:: \mathrm{nbdtrc}(k, n, p) = I_{1 - p}(k + 1, n). Wrapper for the Cephes [1]_ routine `nbdtrc`. The negative binomial distribution is also available as `scipy.stats.nbinom`. Using `nbdtrc` directly can improve performance compared to the ``sf`` method of `scipy.stats.nbinom` (see last example). References ---------- .. [1] Cephes Mathematical Functions Library, http://www.netlib.org/cephes/ Examples -------- Compute the function for ``k=10`` and ``n=5`` at ``p=0.5``. >>> import numpy as np >>> from scipy.special import nbdtrc >>> nbdtrc(10, 5, 0.5) 0.059234619140624986 Compute the function for ``n=10`` and ``p=0.5`` at several points by providing a NumPy array or list for `k`. >>> nbdtrc([5, 10, 15], 10, 0.5) array([0.84912109, 0.41190147, 0.11476147]) Plot the function for four different parameter sets. >>> import matplotlib.pyplot as plt >>> k = np.arange(130) >>> n_parameters = [20, 20, 20, 80] >>> p_parameters = [0.2, 0.5, 0.8, 0.5] >>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot'] >>> parameters_list = list(zip(p_parameters, n_parameters, ... linestyles)) >>> fig, ax = plt.subplots(figsize=(8, 8)) >>> for parameter_set in parameters_list: ... p, n, style = parameter_set ... nbdtrc_vals = nbdtrc(k, n, p) ... ax.plot(k, nbdtrc_vals, label=rf"$n={n},\, p={p}$", ... ls=style) >>> ax.legend() >>> ax.set_xlabel("$k$") >>> ax.set_title("Negative binomial distribution survival function") >>> plt.show() The negative binomial distribution is also available as `scipy.stats.nbinom`. Using `nbdtrc` directly can be much faster than calling the ``sf`` method of `scipy.stats.nbinom`, especially for small arrays or individual values. To get the same results one must use the following parametrization: ``nbinom(n, p).sf(k)=nbdtrc(k, n, p)``. >>> from scipy.stats import nbinom >>> k, n, p = 3, 5, 0.5 >>> nbdtr_res = nbdtrc(k, n, p) # this will often be faster than below >>> stats_res = nbinom(n, p).sf(k) >>> stats_res, nbdtr_res # test that results are equal (0.6367187499999999, 0.6367187499999999) `nbdtrc` can evaluate different parameter sets by providing arrays with shapes compatible for broadcasting for `k`, `n` and `p`. Here we compute the function for three different `k` at four locations `p`, resulting in a 3x4 array. >>> k = np.array([[5], [10], [15]]) >>> p = np.array([0.3, 0.5, 0.7, 0.9]) >>> k.shape, p.shape ((3, 1), (4,)) >>> nbdtrc(k, 5, p) array([[8.49731667e-01, 3.76953125e-01, 4.73489874e-02, 1.46902600e-04], [5.15491059e-01, 5.92346191e-02, 6.72234070e-04, 9.29610100e-09], [2.37507779e-01, 5.90896606e-03, 5.55025308e-06, 3.26346760e-13]])nbdtrinbdtri(k, n, y, out=None) Returns the inverse with respect to the parameter `p` of `y = nbdtr(k, n, p)`, the negative binomial cumulative distribution function. Parameters ---------- k : array_like The maximum number of allowed failures (nonnegative int). n : array_like The target number of successes (positive int). y : array_like The probability of `k` or fewer failures before `n` successes (float). out : ndarray, optional Optional output array for the function results Returns ------- p : scalar or ndarray Probability of success in a single event (float) such that `nbdtr(k, n, p) = y`. See also -------- nbdtr : Cumulative distribution function of the negative binomial. nbdtrc : Negative binomial survival function. scipy.stats.nbinom : negative binomial distribution. nbdtrik : Inverse with respect to `k` of `nbdtr(k, n, p)`. nbdtrin : Inverse with respect to `n` of `nbdtr(k, n, p)`. scipy.stats.nbinom : Negative binomial distribution Notes ----- Wrapper for the Cephes [1]_ routine `nbdtri`. The negative binomial distribution is also available as `scipy.stats.nbinom`. Using `nbdtri` directly can improve performance compared to the ``ppf`` method of `scipy.stats.nbinom`. References ---------- .. [1] Cephes Mathematical Functions Library, http://www.netlib.org/cephes/ Examples -------- `nbdtri` is the inverse of `nbdtr` with respect to `p`. Up to floating point errors the following holds: ``nbdtri(k, n, nbdtr(k, n, p))=p``. >>> import numpy as np >>> from scipy.special import nbdtri, nbdtr >>> k, n, y = 5, 10, 0.2 >>> cdf_val = nbdtr(k, n, y) >>> nbdtri(k, n, cdf_val) 0.20000000000000004 Compute the function for ``k=10`` and ``n=5`` at several points by providing a NumPy array or list for `y`. >>> y = np.array([0.1, 0.4, 0.8]) >>> nbdtri(3, 5, y) array([0.34462319, 0.51653095, 0.69677416]) Plot the function for three different parameter sets. >>> import matplotlib.pyplot as plt >>> n_parameters = [5, 20, 30, 30] >>> k_parameters = [20, 20, 60, 80] >>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot'] >>> parameters_list = list(zip(n_parameters, k_parameters, linestyles)) >>> cdf_vals = np.linspace(0, 1, 1000) >>> fig, ax = plt.subplots(figsize=(8, 8)) >>> for parameter_set in parameters_list: ... n, k, style = parameter_set ... nbdtri_vals = nbdtri(k, n, cdf_vals) ... ax.plot(cdf_vals, nbdtri_vals, label=rf"$k={k},\ n={n}$", ... ls=style) >>> ax.legend() >>> ax.set_ylabel("$p$") >>> ax.set_xlabel("$CDF$") >>> title = "nbdtri: inverse of negative binomial CDF with respect to $p$" >>> ax.set_title(title) >>> plt.show() `nbdtri` can evaluate different parameter sets by providing arrays with shapes compatible for broadcasting for `k`, `n` and `p`. Here we compute the function for three different `k` at four locations `p`, resulting in a 3x4 array. >>> k = np.array([[5], [10], [15]]) >>> y = np.array([0.3, 0.5, 0.7, 0.9]) >>> k.shape, y.shape ((3, 1), (4,)) >>> nbdtri(k, 5, y) array([[0.37258157, 0.45169416, 0.53249956, 0.64578407], [0.24588501, 0.30451981, 0.36778453, 0.46397088], [0.18362101, 0.22966758, 0.28054743, 0.36066188]])nbdtriknbdtrik(y, n, p, out=None) Negative binomial percentile function. Returns the inverse with respect to the parameter `k` of `y = nbdtr(k, n, p)`, the negative binomial cumulative distribution function. Parameters ---------- y : array_like The probability of `k` or fewer failures before `n` successes (float). n : array_like The target number of successes (positive int). p : array_like Probability of success in a single event (float). out : ndarray, optional Optional output array for the function results Returns ------- k : scalar or ndarray The maximum number of allowed failures such that `nbdtr(k, n, p) = y`. See also -------- nbdtr : Cumulative distribution function of the negative binomial. nbdtrc : Survival function of the negative binomial. nbdtri : Inverse with respect to `p` of `nbdtr(k, n, p)`. nbdtrin : Inverse with respect to `n` of `nbdtr(k, n, p)`. scipy.stats.nbinom : Negative binomial distribution Notes ----- Wrapper for the CDFLIB [1]_ Fortran routine `cdfnbn`. Formula 26.5.26 of [2]_, .. math:: \sum_{j=k + 1}^\infty {{n + j - 1}\choose{j}} p^n (1 - p)^j = I_{1 - p}(k + 1, n), is used to reduce calculation of the cumulative distribution function to that of a regularized incomplete beta :math:`I`. Computation of `k` involves a search for a value that produces the desired value of `y`. The search relies on the monotonicity of `y` with `k`. References ---------- .. [1] Barry Brown, James Lovato, and Kathy Russell, CDFLIB: Library of Fortran Routines for Cumulative Distribution Functions, Inverses, and Other Parameters. .. [2] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. Examples -------- Compute the negative binomial cumulative distribution function for an exemplary parameter set. >>> import numpy as np >>> from scipy.special import nbdtr, nbdtrik >>> k, n, p = 5, 2, 0.5 >>> cdf_value = nbdtr(k, n, p) >>> cdf_value 0.9375 Verify that `nbdtrik` recovers the original value for `k`. >>> nbdtrik(cdf_value, n, p) 5.0 Plot the function for different parameter sets. >>> import matplotlib.pyplot as plt >>> p_parameters = [0.2, 0.5, 0.7, 0.5] >>> n_parameters = [30, 30, 30, 80] >>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot'] >>> parameters_list = list(zip(p_parameters, n_parameters, linestyles)) >>> cdf_vals = np.linspace(0, 1, 1000) >>> fig, ax = plt.subplots(figsize=(8, 8)) >>> for parameter_set in parameters_list: ... p, n, style = parameter_set ... nbdtrik_vals = nbdtrik(cdf_vals, n, p) ... ax.plot(cdf_vals, nbdtrik_vals, label=rf"$n={n},\ p={p}$", ... ls=style) >>> ax.legend() >>> ax.set_ylabel("$k$") >>> ax.set_xlabel("$CDF$") >>> ax.set_title("Negative binomial percentile function") >>> plt.show() The negative binomial distribution is also available as `scipy.stats.nbinom`. The percentile function method ``ppf`` returns the result of `nbdtrik` rounded up to integers: >>> from scipy.stats import nbinom >>> q, n, p = 0.6, 5, 0.5 >>> nbinom.ppf(q, n, p), nbdtrik(q, n, p) (5.0, 4.800428460273882)nbdtrinnbdtrin(k, y, p, out=None) Inverse of `nbdtr` vs `n`. Returns the inverse with respect to the parameter `n` of `y = nbdtr(k, n, p)`, the negative binomial cumulative distribution function. Parameters ---------- k : array_like The maximum number of allowed failures (nonnegative int). y : array_like The probability of `k` or fewer failures before `n` successes (float). p : array_like Probability of success in a single event (float). out : ndarray, optional Optional output array for the function results Returns ------- n : scalar or ndarray The number of successes `n` such that `nbdtr(k, n, p) = y`. See also -------- nbdtr : Cumulative distribution function of the negative binomial. nbdtri : Inverse with respect to `p` of `nbdtr(k, n, p)`. nbdtrik : Inverse with respect to `k` of `nbdtr(k, n, p)`. Notes ----- Wrapper for the CDFLIB [1]_ Fortran routine `cdfnbn`. Formula 26.5.26 of [2]_, .. math:: \sum_{j=k + 1}^\infty {{n + j - 1}\choose{j}} p^n (1 - p)^j = I_{1 - p}(k + 1, n), is used to reduce calculation of the cumulative distribution function to that of a regularized incomplete beta :math:`I`. Computation of `n` involves a search for a value that produces the desired value of `y`. The search relies on the monotonicity of `y` with `n`. References ---------- .. [1] Barry Brown, James Lovato, and Kathy Russell, CDFLIB: Library of Fortran Routines for Cumulative Distribution Functions, Inverses, and Other Parameters. .. [2] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. Examples -------- Compute the negative binomial cumulative distribution function for an exemplary parameter set. >>> from scipy.special import nbdtr, nbdtrin >>> k, n, p = 5, 2, 0.5 >>> cdf_value = nbdtr(k, n, p) >>> cdf_value 0.9375 Verify that `nbdtrin` recovers the original value for `n` up to floating point accuracy. >>> nbdtrin(k, cdf_value, p) 1.999999999998137ncfdtrncfdtr(dfn, dfd, nc, f, out=None) Cumulative distribution function of the non-central F distribution. The non-central F describes the distribution of, .. math:: Z = \frac{X/d_n}{Y/d_d} where :math:`X` and :math:`Y` are independently distributed, with :math:`X` distributed non-central :math:`\chi^2` with noncentrality parameter `nc` and :math:`d_n` degrees of freedom, and :math:`Y` distributed :math:`\chi^2` with :math:`d_d` degrees of freedom. Parameters ---------- dfn : array_like Degrees of freedom of the numerator sum of squares. Range (0, inf). dfd : array_like Degrees of freedom of the denominator sum of squares. Range (0, inf). nc : array_like Noncentrality parameter. Should be in range (0, 1e4). f : array_like Quantiles, i.e. the upper limit of integration. out : ndarray, optional Optional output array for the function results Returns ------- cdf : scalar or ndarray The calculated CDF. If all inputs are scalar, the return will be a float. Otherwise it will be an array. See Also -------- ncfdtri : Quantile function; inverse of `ncfdtr` with respect to `f`. ncfdtridfd : Inverse of `ncfdtr` with respect to `dfd`. ncfdtridfn : Inverse of `ncfdtr` with respect to `dfn`. ncfdtrinc : Inverse of `ncfdtr` with respect to `nc`. Notes ----- Wrapper for the CDFLIB [1]_ Fortran routine `cdffnc`. The cumulative distribution function is computed using Formula 26.6.20 of [2]_: .. math:: F(d_n, d_d, n_c, f) = \sum_{j=0}^\infty e^{-n_c/2} \frac{(n_c/2)^j}{j!} I_{x}(\frac{d_n}{2} + j, \frac{d_d}{2}), where :math:`I` is the regularized incomplete beta function, and :math:`x = f d_n/(f d_n + d_d)`. The computation time required for this routine is proportional to the noncentrality parameter `nc`. Very large values of this parameter can consume immense computer resources. This is why the search range is bounded by 10,000. References ---------- .. [1] Barry Brown, James Lovato, and Kathy Russell, CDFLIB: Library of Fortran Routines for Cumulative Distribution Functions, Inverses, and Other Parameters. .. [2] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. Examples -------- >>> import numpy as np >>> from scipy import special >>> from scipy import stats >>> import matplotlib.pyplot as plt Plot the CDF of the non-central F distribution, for nc=0. Compare with the F-distribution from scipy.stats: >>> x = np.linspace(-1, 8, num=500) >>> dfn = 3 >>> dfd = 2 >>> ncf_stats = stats.f.cdf(x, dfn, dfd) >>> ncf_special = special.ncfdtr(dfn, dfd, 0, x) >>> fig = plt.figure() >>> ax = fig.add_subplot(111) >>> ax.plot(x, ncf_stats, 'b-', lw=3) >>> ax.plot(x, ncf_special, 'r-') >>> plt.show()ncfdtrincfdtri(dfn, dfd, nc, p, out=None) Inverse with respect to `f` of the CDF of the non-central F distribution. See `ncfdtr` for more details. Parameters ---------- dfn : array_like Degrees of freedom of the numerator sum of squares. Range (0, inf). dfd : array_like Degrees of freedom of the denominator sum of squares. Range (0, inf). nc : array_like Noncentrality parameter. Should be in range (0, 1e4). p : array_like Value of the cumulative distribution function. Must be in the range [0, 1]. out : ndarray, optional Optional output array for the function results Returns ------- f : scalar or ndarray Quantiles, i.e., the upper limit of integration. See Also -------- ncfdtr : CDF of the non-central F distribution. ncfdtridfd : Inverse of `ncfdtr` with respect to `dfd`. ncfdtridfn : Inverse of `ncfdtr` with respect to `dfn`. ncfdtrinc : Inverse of `ncfdtr` with respect to `nc`. Examples -------- >>> from scipy.special import ncfdtr, ncfdtri Compute the CDF for several values of `f`: >>> f = [0.5, 1, 1.5] >>> p = ncfdtr(2, 3, 1.5, f) >>> p array([ 0.20782291, 0.36107392, 0.47345752]) Compute the inverse. We recover the values of `f`, as expected: >>> ncfdtri(2, 3, 1.5, p) array([ 0.5, 1. , 1.5])ncfdtridfdncfdtridfd(dfn, p, nc, f, out=None) Calculate degrees of freedom (denominator) for the noncentral F-distribution. This is the inverse with respect to `dfd` of `ncfdtr`. See `ncfdtr` for more details. Parameters ---------- dfn : array_like Degrees of freedom of the numerator sum of squares. Range (0, inf). p : array_like Value of the cumulative distribution function. Must be in the range [0, 1]. nc : array_like Noncentrality parameter. Should be in range (0, 1e4). f : array_like Quantiles, i.e., the upper limit of integration. out : ndarray, optional Optional output array for the function results Returns ------- dfd : scalar or ndarray Degrees of freedom of the denominator sum of squares. See Also -------- ncfdtr : CDF of the non-central F distribution. ncfdtri : Quantile function; inverse of `ncfdtr` with respect to `f`. ncfdtridfn : Inverse of `ncfdtr` with respect to `dfn`. ncfdtrinc : Inverse of `ncfdtr` with respect to `nc`. Notes ----- The value of the cumulative noncentral F distribution is not necessarily monotone in either degrees of freedom. There thus may be two values that provide a given CDF value. This routine assumes monotonicity and will find an arbitrary one of the two values. Examples -------- >>> from scipy.special import ncfdtr, ncfdtridfd Compute the CDF for several values of `dfd`: >>> dfd = [1, 2, 3] >>> p = ncfdtr(2, dfd, 0.25, 15) >>> p array([ 0.8097138 , 0.93020416, 0.96787852]) Compute the inverse. We recover the values of `dfd`, as expected: >>> ncfdtridfd(2, p, 0.25, 15) array([ 1., 2., 3.])ncfdtridfnncfdtridfn(p, dfd, nc, f, out=None) Calculate degrees of freedom (numerator) for the noncentral F-distribution. This is the inverse with respect to `dfn` of `ncfdtr`. See `ncfdtr` for more details. Parameters ---------- p : array_like Value of the cumulative distribution function. Must be in the range [0, 1]. dfd : array_like Degrees of freedom of the denominator sum of squares. Range (0, inf). nc : array_like Noncentrality parameter. Should be in range (0, 1e4). f : float Quantiles, i.e., the upper limit of integration. out : ndarray, optional Optional output array for the function results Returns ------- dfn : scalar or ndarray Degrees of freedom of the numerator sum of squares. See Also -------- ncfdtr : CDF of the non-central F distribution. ncfdtri : Quantile function; inverse of `ncfdtr` with respect to `f`. ncfdtridfd : Inverse of `ncfdtr` with respect to `dfd`. ncfdtrinc : Inverse of `ncfdtr` with respect to `nc`. Notes ----- The value of the cumulative noncentral F distribution is not necessarily monotone in either degrees of freedom. There thus may be two values that provide a given CDF value. This routine assumes monotonicity and will find an arbitrary one of the two values. Examples -------- >>> from scipy.special import ncfdtr, ncfdtridfn Compute the CDF for several values of `dfn`: >>> dfn = [1, 2, 3] >>> p = ncfdtr(dfn, 2, 0.25, 15) >>> p array([ 0.92562363, 0.93020416, 0.93188394]) Compute the inverse. We recover the values of `dfn`, as expected: >>> ncfdtridfn(p, 2, 0.25, 15) array([ 1., 2., 3.])ncfdtrincncfdtrinc(dfn, dfd, p, f, out=None) Calculate non-centrality parameter for non-central F distribution. This is the inverse with respect to `nc` of `ncfdtr`. See `ncfdtr` for more details. Parameters ---------- dfn : array_like Degrees of freedom of the numerator sum of squares. Range (0, inf). dfd : array_like Degrees of freedom of the denominator sum of squares. Range (0, inf). p : array_like Value of the cumulative distribution function. Must be in the range [0, 1]. f : array_like Quantiles, i.e., the upper limit of integration. out : ndarray, optional Optional output array for the function results Returns ------- nc : scalar or ndarray Noncentrality parameter. See Also -------- ncfdtr : CDF of the non-central F distribution. ncfdtri : Quantile function; inverse of `ncfdtr` with respect to `f`. ncfdtridfd : Inverse of `ncfdtr` with respect to `dfd`. ncfdtridfn : Inverse of `ncfdtr` with respect to `dfn`. Examples -------- >>> from scipy.special import ncfdtr, ncfdtrinc Compute the CDF for several values of `nc`: >>> nc = [0.5, 1.5, 2.0] >>> p = ncfdtr(2, 3, nc, 15) >>> p array([ 0.96309246, 0.94327955, 0.93304098]) Compute the inverse. We recover the values of `nc`, as expected: >>> ncfdtrinc(2, 3, p, 15) array([ 0.5, 1.5, 2. ])nctdtrnctdtr(df, nc, t, out=None) Cumulative distribution function of the non-central `t` distribution. Parameters ---------- df : array_like Degrees of freedom of the distribution. Should be in range (0, inf). nc : array_like Noncentrality parameter. Should be in range (-1e6, 1e6). t : array_like Quantiles, i.e., the upper limit of integration. out : ndarray, optional Optional output array for the function results Returns ------- cdf : scalar or ndarray The calculated CDF. If all inputs are scalar, the return will be a float. Otherwise, it will be an array. See Also -------- nctdtrit : Inverse CDF (iCDF) of the non-central t distribution. nctdtridf : Calculate degrees of freedom, given CDF and iCDF values. nctdtrinc : Calculate non-centrality parameter, given CDF iCDF values. Examples -------- >>> import numpy as np >>> from scipy import special >>> from scipy import stats >>> import matplotlib.pyplot as plt Plot the CDF of the non-central t distribution, for nc=0. Compare with the t-distribution from scipy.stats: >>> x = np.linspace(-5, 5, num=500) >>> df = 3 >>> nct_stats = stats.t.cdf(x, df) >>> nct_special = special.nctdtr(df, 0, x) >>> fig = plt.figure() >>> ax = fig.add_subplot(111) >>> ax.plot(x, nct_stats, 'b-', lw=3) >>> ax.plot(x, nct_special, 'r-') >>> plt.show()nctdtridfnctdtridf(p, nc, t, out=None) Calculate degrees of freedom for non-central t distribution. See `nctdtr` for more details. Parameters ---------- p : array_like CDF values, in range (0, 1]. nc : array_like Noncentrality parameter. Should be in range (-1e6, 1e6). t : array_like Quantiles, i.e., the upper limit of integration. out : ndarray, optional Optional output array for the function results Returns ------- cdf : scalar or ndarray The calculated CDF. If all inputs are scalar, the return will be a float. Otherwise, it will be an array. See Also -------- nctdtr : CDF of the non-central `t` distribution. nctdtrit : Inverse CDF (iCDF) of the non-central t distribution. nctdtrinc : Calculate non-centrality parameter, given CDF iCDF values.nctdtrincnctdtrinc(df, p, t, out=None) Calculate non-centrality parameter for non-central t distribution. See `nctdtr` for more details. Parameters ---------- df : array_like Degrees of freedom of the distribution. Should be in range (0, inf). p : array_like CDF values, in range (0, 1]. t : array_like Quantiles, i.e., the upper limit of integration. out : ndarray, optional Optional output array for the function results Returns ------- nc : scalar or ndarray Noncentrality parameter See Also -------- nctdtr : CDF of the non-central `t` distribution. nctdtrit : Inverse CDF (iCDF) of the non-central t distribution. nctdtridf : Calculate degrees of freedom, given CDF and iCDF values.nctdtritnctdtrit(df, nc, p, out=None) Inverse cumulative distribution function of the non-central t distribution. See `nctdtr` for more details. Parameters ---------- df : array_like Degrees of freedom of the distribution. Should be in range (0, inf). nc : array_like Noncentrality parameter. Should be in range (-1e6, 1e6). p : array_like CDF values, in range (0, 1]. out : ndarray, optional Optional output array for the function results Returns ------- t : scalar or ndarray Quantiles See Also -------- nctdtr : CDF of the non-central `t` distribution. nctdtridf : Calculate degrees of freedom, given CDF and iCDF values. nctdtrinc : Calculate non-centrality parameter, given CDF iCDF values.ndtrndtr(x, out=None) Cumulative distribution of the standard normal distribution. Returns the area under the standard Gaussian probability density function, integrated from minus infinity to `x` .. math:: \frac{1}{\sqrt{2\pi}} \int_{-\infty}^x \exp(-t^2/2) dt Parameters ---------- x : array_like, real or complex Argument out : ndarray, optional Optional output array for the function results Returns ------- scalar or ndarray The value of the normal CDF evaluated at `x` See Also -------- log_ndtr : Logarithm of ndtr ndtri : Inverse of ndtr, standard normal percentile function erf : Error function erfc : 1 - erf scipy.stats.norm : Normal distribution Examples -------- Evaluate `ndtr` at one point. >>> import numpy as np >>> from scipy.special import ndtr >>> ndtr(0.5) 0.6914624612740131 Evaluate the function at several points by providing a NumPy array or list for `x`. >>> ndtr([0, 0.5, 2]) array([0.5 , 0.69146246, 0.97724987]) Plot the function. >>> import matplotlib.pyplot as plt >>> x = np.linspace(-5, 5, 100) >>> fig, ax = plt.subplots() >>> ax.plot(x, ndtr(x)) >>> ax.set_title("Standard normal cumulative distribution function $\Phi$") >>> plt.show()ndtrindtri(y, out=None) Inverse of `ndtr` vs x Returns the argument x for which the area under the standard normal probability density function (integrated from minus infinity to `x`) is equal to y. Parameters ---------- p : array_like Probability out : ndarray, optional Optional output array for the function results Returns ------- x : scalar or ndarray Value of x such that ``ndtr(x) == p``. See Also -------- ndtr : Standard normal cumulative probability distribution ndtri_exp : Inverse of log_ndtr Examples -------- `ndtri` is the percentile function of the standard normal distribution. This means it returns the inverse of the cumulative density `ndtr`. First, let us compute a cumulative density value. >>> import numpy as np >>> from scipy.special import ndtri, ndtr >>> cdf_val = ndtr(2) >>> cdf_val 0.9772498680518208 Verify that `ndtri` yields the original value for `x` up to floating point errors. >>> ndtri(cdf_val) 2.0000000000000004 Plot the function. For that purpose, we provide a NumPy array as argument. >>> import matplotlib.pyplot as plt >>> x = np.linspace(0.01, 1, 200) >>> fig, ax = plt.subplots() >>> ax.plot(x, ndtri(x)) >>> ax.set_title("Standard normal percentile function") >>> plt.show()ndtri_expndtri_exp(y, out=None) Inverse of `log_ndtr` vs x. Allows for greater precision than `ndtri` composed with `numpy.exp` for very small values of y and for y close to 0. Parameters ---------- y : array_like of float Function argument out : ndarray, optional Optional output array for the function results Returns ------- scalar or ndarray Inverse of the log CDF of the standard normal distribution, evaluated at y. Examples -------- >>> import numpy as np >>> import scipy.special as sc `ndtri_exp` agrees with the naive implementation when the latter does not suffer from underflow. >>> sc.ndtri_exp(-1) -0.33747496376420244 >>> sc.ndtri(np.exp(-1)) -0.33747496376420244 For extreme values of y, the naive approach fails >>> sc.ndtri(np.exp(-800)) -inf >>> sc.ndtri(np.exp(-1e-20)) inf whereas `ndtri_exp` is still able to compute the result to high precision. >>> sc.ndtri_exp(-800) -39.88469483825668 >>> sc.ndtri_exp(-1e-20) 9.262340089798409 See Also -------- log_ndtr, ndtri, ndtrnrdtrimnnrdtrimn(p, x, std, out=None) Calculate mean of normal distribution given other params. Parameters ---------- p : array_like CDF values, in range (0, 1]. x : array_like Quantiles, i.e. the upper limit of integration. std : array_like Standard deviation. out : ndarray, optional Optional output array for the function results Returns ------- mn : scalar or ndarray The mean of the normal distribution. See Also -------- nrdtrimn, ndtrnrdtrisdnrdtrisd(p, x, mn, out=None) Calculate standard deviation of normal distribution given other params. Parameters ---------- p : array_like CDF values, in range (0, 1]. x : array_like Quantiles, i.e. the upper limit of integration. mn : scalar or ndarray The mean of the normal distribution. out : ndarray, optional Optional output array for the function results Returns ------- std : scalar or ndarray Standard deviation. See Also -------- ndtrobl_ang1obl_ang1(m, n, c, x, out=None) Oblate spheroidal angular function of the first kind and its derivative Computes the oblate spheroidal angular function of the first kind and its derivative (with respect to `x`) for mode parameters m>=0 and n>=m, spheroidal parameter `c` and ``|x| < 1.0``. Parameters ---------- m : array_like Mode parameter m (nonnegative) n : array_like Mode parameter n (>= m) c : array_like Spheroidal parameter x : array_like Parameter x (``|x| < 1.0``) out : ndarray, optional Optional output array for the function results Returns ------- s : scalar or ndarray Value of the function sp : scalar or ndarray Value of the derivative vs x See Also -------- obl_ang1_cvobl_ang1_cvobl_ang1_cv(m, n, c, cv, x, out=None) Oblate spheroidal angular function obl_ang1 for precomputed characteristic value Computes the oblate spheroidal angular function of the first kind and its derivative (with respect to `x`) for mode parameters m>=0 and n>=m, spheroidal parameter `c` and ``|x| < 1.0``. Requires pre-computed characteristic value. Parameters ---------- m : array_like Mode parameter m (nonnegative) n : array_like Mode parameter n (>= m) c : array_like Spheroidal parameter cv : array_like Characteristic value x : array_like Parameter x (``|x| < 1.0``) out : ndarray, optional Optional output array for the function results Returns ------- s : scalar or ndarray Value of the function sp : scalar or ndarray Value of the derivative vs x See Also -------- obl_ang1obl_cvobl_cv(m, n, c, out=None) Characteristic value of oblate spheroidal function Computes the characteristic value of oblate spheroidal wave functions of order `m`, `n` (n>=m) and spheroidal parameter `c`. Parameters ---------- m : array_like Mode parameter m (nonnegative) n : array_like Mode parameter n (>= m) c : array_like Spheroidal parameter out : ndarray, optional Optional output array for the function results Returns ------- cv : scalar or ndarray Characteristic valueobl_rad1(m, n, c, x, out=None) Oblate spheroidal radial function of the first kind and its derivative Computes the oblate spheroidal radial function of the first kind and its derivative (with respect to `x`) for mode parameters m>=0 and n>=m, spheroidal parameter `c` and ``|x| < 1.0``. Parameters ---------- m : array_like Mode parameter m (nonnegative) n : array_like Mode parameter n (>= m) c : array_like Spheroidal parameter x : array_like Parameter x (``|x| < 1.0``) out : ndarray, optional Optional output array for the function results Returns ------- s : scalar or ndarray Value of the function sp : scalar or ndarray Value of the derivative vs x See Also -------- obl_rad1_cvobl_rad1_cvobl_rad1_cv(m, n, c, cv, x, out=None) Oblate spheroidal radial function obl_rad1 for precomputed characteristic value Computes the oblate spheroidal radial function of the first kind and its derivative (with respect to `x`) for mode parameters m>=0 and n>=m, spheroidal parameter `c` and ``|x| < 1.0``. Requires pre-computed characteristic value. Parameters ---------- m : array_like Mode parameter m (nonnegative) n : array_like Mode parameter n (>= m) c : array_like Spheroidal parameter cv : array_like Characteristic value x : array_like Parameter x (``|x| < 1.0``) out : ndarray, optional Optional output array for the function results Returns ------- s : scalar or ndarray Value of the function sp : scalar or ndarray Value of the derivative vs x See Also -------- obl_rad1obl_rad2obl_rad2(m, n, c, x, out=None) Oblate spheroidal radial function of the second kind and its derivative. Computes the oblate spheroidal radial function of the second kind and its derivative (with respect to `x`) for mode parameters m>=0 and n>=m, spheroidal parameter `c` and ``|x| < 1.0``. Parameters ---------- m : array_like Mode parameter m (nonnegative) n : array_like Mode parameter n (>= m) c : array_like Spheroidal parameter x : array_like Parameter x (``|x| < 1.0``) out : ndarray, optional Optional output array for the function results Returns ------- s : scalar or ndarray Value of the function sp : scalar or ndarray Value of the derivative vs x See Also -------- obl_rad2_cvobl_rad2_cvobl_rad2_cv(m, n, c, cv, x, out=None) Oblate spheroidal radial function obl_rad2 for precomputed characteristic value Computes the oblate spheroidal radial function of the second kind and its derivative (with respect to `x`) for mode parameters m>=0 and n>=m, spheroidal parameter `c` and ``|x| < 1.0``. Requires pre-computed characteristic value. Parameters ---------- m : array_like Mode parameter m (nonnegative) n : array_like Mode parameter n (>= m) c : array_like Spheroidal parameter cv : array_like Characteristic value x : array_like Parameter x (``|x| < 1.0``) out : ndarray, optional Optional output array for the function results Returns ------- s : scalar or ndarray Value of the function sp : scalar or ndarray Value of the derivative vs x See Also -------- obl_rad2owens_towens_t(h, a, out=None) Owen's T Function. The function T(h, a) gives the probability of the event (X > h and 0 < Y < a * X) where X and Y are independent standard normal random variables. Parameters ---------- h: array_like Input value. a: array_like Input value. out : ndarray, optional Optional output array for the function results Returns ------- t: scalar or ndarray Probability of the event (X > h and 0 < Y < a * X), where X and Y are independent standard normal random variables. Examples -------- >>> from scipy import special >>> a = 3.5 >>> h = 0.78 >>> special.owens_t(h, a) 0.10877216734852274 References ---------- .. [1] M. Patefield and D. Tandy, "Fast and accurate calculation of Owen's T Function", Statistical Software vol. 5, pp. 1-25, 2000.pbdvpbdv(v, x, out=None) Parabolic cylinder function D Returns (d, dp) the parabolic cylinder function Dv(x) in d and the derivative, Dv'(x) in dp. Parameters ---------- v : array_like Real parameter x : array_like Real argument out : ndarray, optional Optional output array for the function results Returns ------- d : scalar or ndarray Value of the function dp : scalar or ndarray Value of the derivative vs xpbvvpbvv(v, x, out=None) Parabolic cylinder function V Returns the parabolic cylinder function Vv(x) in v and the derivative, Vv'(x) in vp. Parameters ---------- v : array_like Real parameter x : array_like Real argument out : ndarray, optional Optional output array for the function results Returns ------- v : scalar or ndarray Value of the function vp : scalar or ndarray Value of the derivative vs xpbwapbwa(a, x, out=None) Parabolic cylinder function W. The function is a particular solution to the differential equation .. math:: y'' + \left(\frac{1}{4}x^2 - a\right)y = 0, for a full definition see section 12.14 in [1]_. Parameters ---------- a : array_like Real parameter x : array_like Real argument out : ndarray, optional Optional output array for the function results Returns ------- w : scalar or ndarray Value of the function wp : scalar or ndarray Value of the derivative in x Notes ----- The function is a wrapper for a Fortran routine by Zhang and Jin [2]_. The implementation is accurate only for ``|a|, |x| < 5`` and returns NaN outside that range. References ---------- .. [1] Digital Library of Mathematical Functions, 14.30. https://dlmf.nist.gov/14.30 .. [2] Zhang, Shanjie and Jin, Jianming. "Computation of Special Functions", John Wiley and Sons, 1996. https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.htmlpdtrpdtr(k, m, out=None) Poisson cumulative distribution function. Defined as the probability that a Poisson-distributed random variable with event rate :math:`m` is less than or equal to :math:`k`. More concretely, this works out to be [1]_ .. math:: \exp(-m) \sum_{j = 0}^{\lfloor{k}\rfloor} \frac{m^j}{j!}. Parameters ---------- k : array_like Number of occurrences (nonnegative, real) m : array_like Shape parameter (nonnegative, real) out : ndarray, optional Optional output array for the function results Returns ------- scalar or ndarray Values of the Poisson cumulative distribution function See Also -------- pdtrc : Poisson survival function pdtrik : inverse of `pdtr` with respect to `k` pdtri : inverse of `pdtr` with respect to `m` References ---------- .. [1] https://en.wikipedia.org/wiki/Poisson_distribution Examples -------- >>> import numpy as np >>> import scipy.special as sc It is a cumulative distribution function, so it converges to 1 monotonically as `k` goes to infinity. >>> sc.pdtr([1, 10, 100, np.inf], 1) array([0.73575888, 0.99999999, 1. , 1. ]) It is discontinuous at integers and constant between integers. >>> sc.pdtr([1, 1.5, 1.9, 2], 1) array([0.73575888, 0.73575888, 0.73575888, 0.9196986 ])pdtrcpdtrc(k, m, out=None) Poisson survival function Returns the sum of the terms from k+1 to infinity of the Poisson distribution: sum(exp(-m) * m**j / j!, j=k+1..inf) = gammainc( k+1, m). Arguments must both be non-negative doubles. Parameters ---------- k : array_like Number of occurrences (nonnegative, real) m : array_like Shape parameter (nonnegative, real) out : ndarray, optional Optional output array for the function results Returns ------- scalar or ndarray Values of the Poisson survival function See Also -------- pdtr : Poisson cumulative distribution function pdtrik : inverse of `pdtr` with respect to `k` pdtri : inverse of `pdtr` with respect to `m`pdtripdtri(k, y, out=None) Inverse to `pdtr` vs m Returns the Poisson variable `m` such that the sum from 0 to `k` of the Poisson density is equal to the given probability `y`: calculated by ``gammaincinv(k + 1, y)``. `k` must be a nonnegative integer and `y` between 0 and 1. Parameters ---------- k : array_like Number of occurrences (nonnegative, real) y : array_like Probability out : ndarray, optional Optional output array for the function results Returns ------- scalar or ndarray Values of the shape paramter `m` such that ``pdtr(k, m) = p`` See Also -------- pdtr : Poisson cumulative distribution function pdtrc : Poisson survival function pdtrik : inverse of `pdtr` with respect to `k`pdtrikpdtrik(p, m, out=None) Inverse to `pdtr` vs `m`. Parameters ---------- m : array_like Shape parameter (nonnegative, real) p : array_like Probability out : ndarray, optional Optional output array for the function results Returns ------- scalar or ndarray The number of occurrences `k` such that ``pdtr(k, m) = p`` See Also -------- pdtr : Poisson cumulative distribution function pdtrc : Poisson survival function pdtri : inverse of `pdtr` with respect to `m`pochpoch(z, m, out=None) Pochhammer symbol. The Pochhammer symbol (rising factorial) is defined as .. math:: (z)_m = \frac{\Gamma(z + m)}{\Gamma(z)} For positive integer `m` it reads .. math:: (z)_m = z (z + 1) ... (z + m - 1) See [dlmf]_ for more details. Parameters ---------- z, m : array_like Real-valued arguments. out : ndarray, optional Optional output array for the function results Returns ------- scalar or ndarray The value of the function. References ---------- .. [dlmf] Nist, Digital Library of Mathematical Functions https://dlmf.nist.gov/5.2#iii Examples -------- >>> import scipy.special as sc It is 1 when m is 0. >>> sc.poch([1, 2, 3, 4], 0) array([1., 1., 1., 1.]) For z equal to 1 it reduces to the factorial function. >>> sc.poch(1, 5) 120.0 >>> 1 * 2 * 3 * 4 * 5 120 It can be expressed in terms of the gamma function. >>> z, m = 3.7, 2.1 >>> sc.poch(z, m) 20.529581933776953 >>> sc.gamma(z + m) / sc.gamma(z) 20.52958193377696powm1powm1(x, y, out=None) Computes ``x**y - 1``. This function is useful when `y` is near 0, or when `x` is near 1. The function is implemented for real types only (unlike ``numpy.power``, which accepts complex inputs). Parameters ---------- x : array_like The base. Must be a real type (i.e. integer or float, not complex). y : array_like The exponent. Must be a real type (i.e. integer or float, not complex). Returns ------- array_like Result of the calculation Notes ----- .. versionadded:: 1.10.0 The underlying code is implemented for single precision and double precision floats only. Unlike `numpy.power`, integer inputs to `powm1` are converted to floating point, and complex inputs are not accepted. Note the following edge cases: * ``powm1(x, 0)`` returns 0 for any ``x``, including 0, ``inf`` and ``nan``. * ``powm1(1, y)`` returns 0 for any ``y``, including ``nan`` and ``inf``. Examples -------- >>> import numpy as np >>> from scipy.special import powm1 >>> x = np.array([1.2, 10.0, 0.9999999975]) >>> y = np.array([1e-9, 1e-11, 0.1875]) >>> powm1(x, y) array([ 1.82321557e-10, 2.30258509e-11, -4.68749998e-10]) It can be verified that the relative errors in those results are less than 2.5e-16. Compare that to the result of ``x**y - 1``, where the relative errors are all larger than 8e-8: >>> x**y - 1 array([ 1.82321491e-10, 2.30258035e-11, -4.68750039e-10])pro_ang1pro_ang1(m, n, c, x, out=None) Prolate spheroidal angular function of the first kind and its derivative Computes the prolate spheroidal angular function of the first kind and its derivative (with respect to `x`) for mode parameters m>=0 and n>=m, spheroidal parameter `c` and ``|x| < 1.0``. Parameters ---------- m : array_like Nonnegative mode parameter m n : array_like Mode parameter n (>= m) c : array_like Spheroidal parameter x : array_like Real parameter (``|x| < 1.0``) out : ndarray, optional Optional output array for the function results Returns ------- s : scalar or ndarray Value of the function sp : scalar or ndarray Value of the derivative vs xpro_ang1_cvpro_ang1_cv(m, n, c, cv, x, out=None) Prolate spheroidal angular function pro_ang1 for precomputed characteristic value Computes the prolate spheroidal angular function of the first kind and its derivative (with respect to `x`) for mode parameters m>=0 and n>=m, spheroidal parameter `c` and ``|x| < 1.0``. Requires pre-computed characteristic value. Parameters ---------- m : array_like Nonnegative mode parameter m n : array_like Mode parameter n (>= m) c : array_like Spheroidal parameter cv : array_like Characteristic value x : array_like Real parameter (``|x| < 1.0``) out : ndarray, optional Optional output array for the function results Returns ------- s : scalar or ndarray Value of the function sp : scalar or ndarray Value of the derivative vs xpro_cvpro_cv(m, n, c, out=None) Characteristic value of prolate spheroidal function Computes the characteristic value of prolate spheroidal wave functions of order `m`, `n` (n>=m) and spheroidal parameter `c`. Parameters ---------- m : array_like Nonnegative mode parameter m n : array_like Mode parameter n (>= m) c : array_like Spheroidal parameter out : ndarray, optional Optional output array for the function results Returns ------- cv : scalar or ndarray Characteristic valuepro_rad1pro_rad1(m, n, c, x, out=None) Prolate spheroidal radial function of the first kind and its derivative Computes the prolate spheroidal radial function of the first kind and its derivative (with respect to `x`) for mode parameters m>=0 and n>=m, spheroidal parameter `c` and ``|x| < 1.0``. Parameters ---------- m : array_like Nonnegative mode parameter m n : array_like Mode parameter n (>= m) c : array_like Spheroidal parameter x : array_like Real parameter (``|x| < 1.0``) out : ndarray, optional Optional output array for the function results Returns ------- s : scalar or ndarray Value of the function sp : scalar or ndarray Value of the derivative vs xpro_rad1_cvpro_rad1_cv(m, n, c, cv, x, out=None) Prolate spheroidal radial function pro_rad1 for precomputed characteristic value Computes the prolate spheroidal radial function of the first kind and its derivative (with respect to `x`) for mode parameters m>=0 and n>=m, spheroidal parameter `c` and ``|x| < 1.0``. Requires pre-computed characteristic value. Parameters ---------- m : array_like Nonnegative mode parameter m n : array_like Mode parameter n (>= m) c : array_like Spheroidal parameter cv : array_like Characteristic value x : array_like Real parameter (``|x| < 1.0``) out : ndarray, optional Optional output array for the function results Returns ------- s : scalar or ndarray Value of the function sp : scalar or ndarray Value of the derivative vs xpro_rad2pro_rad2(m, n, c, x, out=None) Prolate spheroidal radial function of the second kind and its derivative Computes the prolate spheroidal radial function of the second kind and its derivative (with respect to `x`) for mode parameters m>=0 and n>=m, spheroidal parameter `c` and ``|x| < 1.0``. Parameters ---------- m : array_like Nonnegative mode parameter m n : array_like Mode parameter n (>= m) c : array_like Spheroidal parameter cv : array_like Characteristic value x : array_like Real parameter (``|x| < 1.0``) out : ndarray, optional Optional output array for the function results Returns ------- s : scalar or ndarray Value of the function sp : scalar or ndarray Value of the derivative vs xpro_rad2_cvpro_rad2_cv(m, n, c, cv, x, out=None) Prolate spheroidal radial function pro_rad2 for precomputed characteristic value Computes the prolate spheroidal radial function of the second kind and its derivative (with respect to `x`) for mode parameters m>=0 and n>=m, spheroidal parameter `c` and ``|x| < 1.0``. Requires pre-computed characteristic value. Parameters ---------- m : array_like Nonnegative mode parameter m n : array_like Mode parameter n (>= m) c : array_like Spheroidal parameter cv : array_like Characteristic value x : array_like Real parameter (``|x| < 1.0``) out : ndarray, optional Optional output array for the function results Returns ------- s : scalar or ndarray Value of the function sp : scalar or ndarray Value of the derivative vs xpseudo_huberpseudo_huber(delta, r, out=None) Pseudo-Huber loss function. .. math:: \mathrm{pseudo\_huber}(\delta, r) = \delta^2 \left( \sqrt{ 1 + \left( \frac{r}{\delta} \right)^2 } - 1 \right) Parameters ---------- delta : array_like Input array, indicating the soft quadratic vs. linear loss changepoint. r : array_like Input array, possibly representing residuals. out : ndarray, optional Optional output array for the function results Returns ------- res : scalar or ndarray The computed Pseudo-Huber loss function values. See also -------- huber: Similar function which this function approximates Notes ----- Like `huber`, `pseudo_huber` often serves as a robust loss function in statistics or machine learning to reduce the influence of outliers. Unlike `huber`, `pseudo_huber` is smooth. Typically, `r` represents residuals, the difference between a model prediction and data. Then, for :math:`|r|\leq\delta`, `pseudo_huber` resembles the squared error and for :math:`|r|>\delta` the absolute error. This way, the Pseudo-Huber loss often achieves a fast convergence in model fitting for small residuals like the squared error loss function and still reduces the influence of outliers (:math:`|r|>\delta`) like the absolute error loss. As :math:`\delta` is the cutoff between squared and absolute error regimes, it has to be tuned carefully for each problem. `pseudo_huber` is also convex, making it suitable for gradient based optimization. [1]_ [2]_ .. versionadded:: 0.15.0 References ---------- .. [1] Hartley, Zisserman, "Multiple View Geometry in Computer Vision". 2003. Cambridge University Press. p. 619 .. [2] Charbonnier et al. "Deterministic edge-preserving regularization in computed imaging". 1997. IEEE Trans. Image Processing. 6 (2): 298 - 311. Examples -------- Import all necessary modules. >>> import numpy as np >>> from scipy.special import pseudo_huber, huber >>> import matplotlib.pyplot as plt Calculate the function for ``delta=1`` at ``r=2``. >>> pseudo_huber(1., 2.) 1.2360679774997898 Calculate the function at ``r=2`` for different `delta` by providing a list or NumPy array for `delta`. >>> pseudo_huber([1., 2., 4.], 3.) array([2.16227766, 3.21110255, 4. ]) Calculate the function for ``delta=1`` at several points by providing a list or NumPy array for `r`. >>> pseudo_huber(2., np.array([1., 1.5, 3., 4.])) array([0.47213595, 1. , 3.21110255, 4.94427191]) The function can be calculated for different `delta` and `r` by providing arrays for both with compatible shapes for broadcasting. >>> r = np.array([1., 2.5, 8., 10.]) >>> deltas = np.array([[1.], [5.], [9.]]) >>> print(r.shape, deltas.shape) (4,) (3, 1) >>> pseudo_huber(deltas, r) array([[ 0.41421356, 1.6925824 , 7.06225775, 9.04987562], [ 0.49509757, 2.95084972, 22.16990566, 30.90169944], [ 0.49846624, 3.06693762, 27.37435121, 40.08261642]]) Plot the function for different `delta`. >>> x = np.linspace(-4, 4, 500) >>> deltas = [1, 2, 3] >>> linestyles = ["dashed", "dotted", "dashdot"] >>> fig, ax = plt.subplots() >>> combined_plot_parameters = list(zip(deltas, linestyles)) >>> for delta, style in combined_plot_parameters: ... ax.plot(x, pseudo_huber(delta, x), label=f"$\delta={delta}$", ... ls=style) >>> ax.legend(loc="upper center") >>> ax.set_xlabel("$x$") >>> ax.set_title("Pseudo-Huber loss function $h_{\delta}(x)$") >>> ax.set_xlim(-4, 4) >>> ax.set_ylim(0, 8) >>> plt.show() Finally, illustrate the difference between `huber` and `pseudo_huber` by plotting them and their gradients with respect to `r`. The plot shows that `pseudo_huber` is continuously differentiable while `huber` is not at the points :math:`\pm\delta`. >>> def huber_grad(delta, x): ... grad = np.copy(x) ... linear_area = np.argwhere(np.abs(x) > delta) ... grad[linear_area]=delta*np.sign(x[linear_area]) ... return grad >>> def pseudo_huber_grad(delta, x): ... return x* (1+(x/delta)**2)**(-0.5) >>> x=np.linspace(-3, 3, 500) >>> delta = 1. >>> fig, ax = plt.subplots(figsize=(7, 7)) >>> ax.plot(x, huber(delta, x), label="Huber", ls="dashed") >>> ax.plot(x, huber_grad(delta, x), label="Huber Gradient", ls="dashdot") >>> ax.plot(x, pseudo_huber(delta, x), label="Pseudo-Huber", ls="dotted") >>> ax.plot(x, pseudo_huber_grad(delta, x), label="Pseudo-Huber Gradient", ... ls="solid") >>> ax.legend(loc="upper center") >>> plt.show()psipsi(z, out=None) The digamma function. The logarithmic derivative of the gamma function evaluated at ``z``. Parameters ---------- z : array_like Real or complex argument. out : ndarray, optional Array for the computed values of ``psi``. Returns ------- digamma : scalar or ndarray Computed values of ``psi``. Notes ----- For large values not close to the negative real axis, ``psi`` is computed using the asymptotic series (5.11.2) from [1]_. For small arguments not close to the negative real axis, the recurrence relation (5.5.2) from [1]_ is used until the argument is large enough to use the asymptotic series. For values close to the negative real axis, the reflection formula (5.5.4) from [1]_ is used first. Note that ``psi`` has a family of zeros on the negative real axis which occur between the poles at nonpositive integers. Around the zeros the reflection formula suffers from cancellation and the implementation loses precision. The sole positive zero and the first negative zero, however, are handled separately by precomputing series expansions using [2]_, so the function should maintain full accuracy around the origin. References ---------- .. [1] NIST Digital Library of Mathematical Functions https://dlmf.nist.gov/5 .. [2] Fredrik Johansson and others. "mpmath: a Python library for arbitrary-precision floating-point arithmetic" (Version 0.19) http://mpmath.org/ Examples -------- >>> from scipy.special import psi >>> z = 3 + 4j >>> psi(z) (1.55035981733341+1.0105022091860445j) Verify psi(z) = psi(z + 1) - 1/z: >>> psi(z + 1) - 1/z (1.55035981733341+1.0105022091860445j)radianradian(d, m, s, out=None) Convert from degrees to radians. Returns the angle given in (d)egrees, (m)inutes, and (s)econds in radians. Parameters ---------- d : array_like Degrees, can be real-valued. m : array_like Minutes, can be real-valued. s : array_like Seconds, can be real-valued. out : ndarray, optional Optional output array for the function results. Returns ------- scalar or ndarray Values of the inputs in radians. Examples -------- >>> import scipy.special as sc There are many ways to specify an angle. >>> sc.radian(90, 0, 0) 1.5707963267948966 >>> sc.radian(0, 60 * 90, 0) 1.5707963267948966 >>> sc.radian(0, 0, 60**2 * 90) 1.5707963267948966 The inputs can be real-valued. >>> sc.radian(1.5, 0, 0) 0.02617993877991494 >>> sc.radian(1, 30, 0) 0.02617993877991494rel_entrrel_entr(x, y, out=None) Elementwise function for computing relative entropy. .. math:: \mathrm{rel\_entr}(x, y) = \begin{cases} x \log(x / y) & x > 0, y > 0 \\ 0 & x = 0, y \ge 0 \\ \infty & \text{otherwise} \end{cases} Parameters ---------- x, y : array_like Input arrays out : ndarray, optional Optional output array for the function results Returns ------- scalar or ndarray Relative entropy of the inputs See Also -------- entr, kl_div, scipy.stats.entropy Notes ----- .. versionadded:: 0.15.0 This function is jointly convex in x and y. The origin of this function is in convex programming; see [1]_. Given two discrete probability distributions :math:`p_1, \ldots, p_n` and :math:`q_1, \ldots, q_n`, the definition of relative entropy in the context of *information theory* is .. math:: \sum_{i = 1}^n \mathrm{rel\_entr}(p_i, q_i). To compute the latter quantity, use `scipy.stats.entropy`. See [2]_ for details. References ---------- .. [1] Boyd, Stephen and Lieven Vandenberghe. *Convex optimization*. Cambridge University Press, 2004. :doi:`https://doi.org/10.1017/CBO9780511804441` .. [2] Kullback-Leibler divergence, https://en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergencergammargamma(z, out=None) Reciprocal of the gamma function. Defined as :math:`1 / \Gamma(z)`, where :math:`\Gamma` is the gamma function. For more on the gamma function see `gamma`. Parameters ---------- z : array_like Real or complex valued input out : ndarray, optional Optional output array for the function results Returns ------- scalar or ndarray Function results Notes ----- The gamma function has no zeros and has simple poles at nonpositive integers, so `rgamma` is an entire function with zeros at the nonpositive integers. See the discussion in [dlmf]_ for more details. See Also -------- gamma, gammaln, loggamma References ---------- .. [dlmf] Nist, Digital Library of Mathematical functions, https://dlmf.nist.gov/5.2#i Examples -------- >>> import scipy.special as sc It is the reciprocal of the gamma function. >>> sc.rgamma([1, 2, 3, 4]) array([1. , 1. , 0.5 , 0.16666667]) >>> 1 / sc.gamma([1, 2, 3, 4]) array([1. , 1. , 0.5 , 0.16666667]) It is zero at nonpositive integers. >>> sc.rgamma([0, -1, -2, -3]) array([0., 0., 0., 0.]) It rapidly underflows to zero along the positive real axis. >>> sc.rgamma([10, 100, 179]) array([2.75573192e-006, 1.07151029e-156, 0.00000000e+000])roundround(x, out=None) Round to the nearest integer. Returns the nearest integer to `x`. If `x` ends in 0.5 exactly, the nearest even integer is chosen. Parameters ---------- x : array_like Real valued input. out : ndarray, optional Optional output array for the function results. Returns ------- scalar or ndarray The nearest integers to the elements of `x`. The result is of floating type, not integer type. Examples -------- >>> import scipy.special as sc It rounds to even. >>> sc.round([0.5, 1.5]) array([0., 2.])shichi(x, out=None) Hyperbolic sine and cosine integrals. The hyperbolic sine integral is .. math:: \int_0^x \frac{\sinh{t}}{t}dt and the hyperbolic cosine integral is .. math:: \gamma + \log(x) + \int_0^x \frac{\cosh{t} - 1}{t} dt where :math:`\gamma` is Euler's constant and :math:`\log` is the principal branch of the logarithm [1]_. Parameters ---------- x : array_like Real or complex points at which to compute the hyperbolic sine and cosine integrals. out : tuple of ndarray, optional Optional output arrays for the function results Returns ------- si : scalar or ndarray Hyperbolic sine integral at ``x`` ci : scalar or ndarray Hyperbolic cosine integral at ``x`` See Also -------- sici : Sine and cosine integrals. exp1 : Exponential integral E1. expi : Exponential integral Ei. Notes ----- For real arguments with ``x < 0``, ``chi`` is the real part of the hyperbolic cosine integral. For such points ``chi(x)`` and ``chi(x + 0j)`` differ by a factor of ``1j*pi``. For real arguments the function is computed by calling Cephes' [2]_ *shichi* routine. For complex arguments the algorithm is based on Mpmath's [3]_ *shi* and *chi* routines. References ---------- .. [1] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. (See Section 5.2.) .. [2] Cephes Mathematical Functions Library, http://www.netlib.org/cephes/ .. [3] Fredrik Johansson and others. "mpmath: a Python library for arbitrary-precision floating-point arithmetic" (Version 0.19) http://mpmath.org/ Examples -------- >>> import numpy as np >>> import matplotlib.pyplot as plt >>> from scipy.special import shichi, sici `shichi` accepts real or complex input: >>> shichi(0.5) (0.5069967498196671, -0.05277684495649357) >>> shichi(0.5 + 2.5j) ((0.11772029666668238+1.831091777729851j), (0.29912435887648825+1.7395351121166562j)) The hyperbolic sine and cosine integrals Shi(z) and Chi(z) are related to the sine and cosine integrals Si(z) and Ci(z) by * Shi(z) = -i*Si(i*z) * Chi(z) = Ci(-i*z) + i*pi/2 >>> z = 0.25 + 5j >>> shi, chi = shichi(z) >>> shi, -1j*sici(1j*z)[0] # Should be the same. ((-0.04834719325101729+1.5469354086921228j), (-0.04834719325101729+1.5469354086921228j)) >>> chi, sici(-1j*z)[1] + 1j*np.pi/2 # Should be the same. ((-0.19568708973868087+1.556276312103824j), (-0.19568708973868087+1.556276312103824j)) Plot the functions evaluated on the real axis: >>> xp = np.geomspace(1e-8, 4.0, 250) >>> x = np.concatenate((-xp[::-1], xp)) >>> shi, chi = shichi(x) >>> fig, ax = plt.subplots() >>> ax.plot(x, shi, label='Shi(x)') >>> ax.plot(x, chi, '--', label='Chi(x)') >>> ax.set_xlabel('x') >>> ax.set_title('Hyperbolic Sine and Cosine Integrals') >>> ax.legend(shadow=True, framealpha=1, loc='lower right') >>> ax.grid(True) >>> plt.show()sici(x, out=None) Sine and cosine integrals. The sine integral is .. math:: \int_0^x \frac{\sin{t}}{t}dt and the cosine integral is .. math:: \gamma + \log(x) + \int_0^x \frac{\cos{t} - 1}{t}dt where :math:`\gamma` is Euler's constant and :math:`\log` is the principal branch of the logarithm [1]_. Parameters ---------- x : array_like Real or complex points at which to compute the sine and cosine integrals. out : tuple of ndarray, optional Optional output arrays for the function results Returns ------- si : scalar or ndarray Sine integral at ``x`` ci : scalar or ndarray Cosine integral at ``x`` See Also -------- shichi : Hyperbolic sine and cosine integrals. exp1 : Exponential integral E1. expi : Exponential integral Ei. Notes ----- For real arguments with ``x < 0``, ``ci`` is the real part of the cosine integral. For such points ``ci(x)`` and ``ci(x + 0j)`` differ by a factor of ``1j*pi``. For real arguments the function is computed by calling Cephes' [2]_ *sici* routine. For complex arguments the algorithm is based on Mpmath's [3]_ *si* and *ci* routines. References ---------- .. [1] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. (See Section 5.2.) .. [2] Cephes Mathematical Functions Library, http://www.netlib.org/cephes/ .. [3] Fredrik Johansson and others. "mpmath: a Python library for arbitrary-precision floating-point arithmetic" (Version 0.19) http://mpmath.org/ Examples -------- >>> import numpy as np >>> import matplotlib.pyplot as plt >>> from scipy.special import sici, exp1 `sici` accepts real or complex input: >>> sici(2.5) (1.7785201734438267, 0.2858711963653835) >>> sici(2.5 + 3j) ((4.505735874563953+0.06863305018999577j), (0.0793644206906966-2.935510262937543j)) For z in the right half plane, the sine and cosine integrals are related to the exponential integral E1 (implemented in SciPy as `scipy.special.exp1`) by * Si(z) = (E1(i*z) - E1(-i*z))/2i + pi/2 * Ci(z) = -(E1(i*z) + E1(-i*z))/2 See [1]_ (equations 5.2.21 and 5.2.23). We can verify these relations: >>> z = 2 - 3j >>> sici(z) ((4.54751388956229-1.3991965806460565j), (1.408292501520851+2.9836177420296055j)) >>> (exp1(1j*z) - exp1(-1j*z))/2j + np.pi/2 # Same as sine integral (4.54751388956229-1.3991965806460565j) >>> -(exp1(1j*z) + exp1(-1j*z))/2 # Same as cosine integral (1.408292501520851+2.9836177420296055j) Plot the functions evaluated on the real axis; the dotted horizontal lines are at pi/2 and -pi/2: >>> x = np.linspace(-16, 16, 150) >>> si, ci = sici(x) >>> fig, ax = plt.subplots() >>> ax.plot(x, si, label='Si(x)') >>> ax.plot(x, ci, '--', label='Ci(x)') >>> ax.legend(shadow=True, framealpha=1, loc='upper left') >>> ax.set_xlabel('x') >>> ax.set_title('Sine and Cosine Integrals') >>> ax.axhline(np.pi/2, linestyle=':', alpha=0.5, color='k') >>> ax.axhline(-np.pi/2, linestyle=':', alpha=0.5, color='k') >>> ax.grid(True) >>> plt.show()sindgsindg(x, out=None) Sine of the angle `x` given in degrees. Parameters ---------- x : array_like Angle, given in degrees. out : ndarray, optional Optional output array for the function results. Returns ------- scalar or ndarray Sine at the input. See Also -------- cosdg, tandg, cotdg Examples -------- >>> import numpy as np >>> import scipy.special as sc It is more accurate than using sine directly. >>> x = 180 * np.arange(3) >>> sc.sindg(x) array([ 0., -0., 0.]) >>> np.sin(x * np.pi / 180) array([ 0.0000000e+00, 1.2246468e-16, -2.4492936e-16])smirnovsmirnov(n, d, out=None) Kolmogorov-Smirnov complementary cumulative distribution function Returns the exact Kolmogorov-Smirnov complementary cumulative distribution function,(aka the Survival Function) of Dn+ (or Dn-) for a one-sided test of equality between an empirical and a theoretical distribution. It is equal to the probability that the maximum difference between a theoretical distribution and an empirical one based on `n` samples is greater than d. Parameters ---------- n : int Number of samples d : float array_like Deviation between the Empirical CDF (ECDF) and the target CDF. out : ndarray, optional Optional output array for the function results Returns ------- scalar or ndarray The value(s) of smirnov(n, d), Prob(Dn+ >= d) (Also Prob(Dn- >= d)) See Also -------- smirnovi : The Inverse Survival Function for the distribution scipy.stats.ksone : Provides the functionality as a continuous distribution kolmogorov, kolmogi : Functions for the two-sided distribution Notes ----- `smirnov` is used by `stats.kstest` in the application of the Kolmogorov-Smirnov Goodness of Fit test. For historial reasons this function is exposed in `scpy.special`, but the recommended way to achieve the most accurate CDF/SF/PDF/PPF/ISF computations is to use the `stats.ksone` distribution. Examples -------- >>> import numpy as np >>> from scipy.special import smirnov >>> from scipy.stats import norm Show the probability of a gap at least as big as 0, 0.5 and 1.0 for a sample of size 5. >>> smirnov(5, [0, 0.5, 1.0]) array([ 1. , 0.056, 0. ]) Compare a sample of size 5 against N(0, 1), the standard normal distribution with mean 0 and standard deviation 1. `x` is the sample. >>> x = np.array([-1.392, -0.135, 0.114, 0.190, 1.82]) >>> target = norm(0, 1) >>> cdfs = target.cdf(x) >>> cdfs array([0.0819612 , 0.44630594, 0.5453811 , 0.57534543, 0.9656205 ]) Construct the empirical CDF and the K-S statistics (Dn+, Dn-, Dn). >>> n = len(x) >>> ecdfs = np.arange(n+1, dtype=float)/n >>> cols = np.column_stack([x, ecdfs[1:], cdfs, cdfs - ecdfs[:n], ... ecdfs[1:] - cdfs]) >>> with np.printoptions(precision=3): ... print(cols) [[-1.392 0.2 0.082 0.082 0.118] [-0.135 0.4 0.446 0.246 -0.046] [ 0.114 0.6 0.545 0.145 0.055] [ 0.19 0.8 0.575 -0.025 0.225] [ 1.82 1. 0.966 0.166 0.034]] >>> gaps = cols[:, -2:] >>> Dnpm = np.max(gaps, axis=0) >>> print(f'Dn-={Dnpm[0]:f}, Dn+={Dnpm[1]:f}') Dn-=0.246306, Dn+=0.224655 >>> probs = smirnov(n, Dnpm) >>> print(f'For a sample of size {n} drawn from N(0, 1):', ... f' Smirnov n={n}: Prob(Dn- >= {Dnpm[0]:f}) = {probs[0]:.4f}', ... f' Smirnov n={n}: Prob(Dn+ >= {Dnpm[1]:f}) = {probs[1]:.4f}', ... sep='\n') For a sample of size 5 drawn from N(0, 1): Smirnov n=5: Prob(Dn- >= 0.246306) = 0.4711 Smirnov n=5: Prob(Dn+ >= 0.224655) = 0.5245 Plot the empirical CDF and the standard normal CDF. >>> import matplotlib.pyplot as plt >>> plt.step(np.concatenate(([-2.5], x, [2.5])), ... np.concatenate((ecdfs, [1])), ... where='post', label='Empirical CDF') >>> xx = np.linspace(-2.5, 2.5, 100) >>> plt.plot(xx, target.cdf(xx), '--', label='CDF for N(0, 1)') Add vertical lines marking Dn+ and Dn-. >>> iminus, iplus = np.argmax(gaps, axis=0) >>> plt.vlines([x[iminus]], ecdfs[iminus], cdfs[iminus], color='r', ... alpha=0.5, lw=4) >>> plt.vlines([x[iplus]], cdfs[iplus], ecdfs[iplus+1], color='m', ... alpha=0.5, lw=4) >>> plt.grid(True) >>> plt.legend(framealpha=1, shadow=True) >>> plt.show()smirnovismirnovi(n, p, out=None) Inverse to `smirnov` Returns `d` such that ``smirnov(n, d) == p``, the critical value corresponding to `p`. Parameters ---------- n : int Number of samples p : float array_like Probability out : ndarray, optional Optional output array for the function results Returns ------- scalar or ndarray The value(s) of smirnovi(n, p), the critical values. See Also -------- smirnov : The Survival Function (SF) for the distribution scipy.stats.ksone : Provides the functionality as a continuous distribution kolmogorov, kolmogi : Functions for the two-sided distribution scipy.stats.kstwobign : Two-sided Kolmogorov-Smirnov distribution, large n Notes ----- `smirnov` is used by `stats.kstest` in the application of the Kolmogorov-Smirnov Goodness of Fit test. For historial reasons this function is exposed in `scpy.special`, but the recommended way to achieve the most accurate CDF/SF/PDF/PPF/ISF computations is to use the `stats.ksone` distribution. Examples -------- >>> from scipy.special import smirnovi, smirnov >>> n = 24 >>> deviations = [0.1, 0.2, 0.3] Use `smirnov` to compute the complementary CDF of the Smirnov distribution for the given number of samples and deviations. >>> p = smirnov(n, deviations) >>> p array([0.58105083, 0.12826832, 0.01032231]) The inverse function ``smirnovi(n, p)`` returns ``deviations``. >>> smirnovi(n, p) array([0.1, 0.2, 0.3])spencespence(z, out=None) Spence's function, also known as the dilogarithm. It is defined to be .. math:: \int_1^z \frac{\log(t)}{1 - t}dt for complex :math:`z`, where the contour of integration is taken to avoid the branch cut of the logarithm. Spence's function is analytic everywhere except the negative real axis where it has a branch cut. Parameters ---------- z : array_like Points at which to evaluate Spence's function out : ndarray, optional Optional output array for the function results Returns ------- s : scalar or ndarray Computed values of Spence's function Notes ----- There is a different convention which defines Spence's function by the integral .. math:: -\int_0^z \frac{\log(1 - t)}{t}dt; this is our ``spence(1 - z)``. Examples -------- >>> import numpy as np >>> from scipy.special import spence >>> import matplotlib.pyplot as plt The function is defined for complex inputs: >>> spence([1-1j, 1.5+2j, 3j, -10-5j]) array([-0.20561676+0.91596559j, -0.86766909-1.39560134j, -0.59422064-2.49129918j, -1.14044398+6.80075924j]) For complex inputs on the branch cut, which is the negative real axis, the function returns the limit for ``z`` with positive imaginary part. For example, in the following, note the sign change of the imaginary part of the output for ``z = -2`` and ``z = -2 - 1e-8j``: >>> spence([-2 + 1e-8j, -2, -2 - 1e-8j]) array([2.32018041-3.45139229j, 2.32018042-3.4513923j , 2.32018041+3.45139229j]) The function returns ``nan`` for real inputs on the branch cut: >>> spence(-1.5) nan Verify some particular values: ``spence(0) = pi**2/6``, ``spence(1) = 0`` and ``spence(2) = -pi**2/12``. >>> spence([0, 1, 2]) array([ 1.64493407, 0. , -0.82246703]) >>> np.pi**2/6, -np.pi**2/12 (1.6449340668482264, -0.8224670334241132) Verify the identity:: spence(z) + spence(1 - z) = pi**2/6 - log(z)*log(1 - z) >>> z = 3 + 4j >>> spence(z) + spence(1 - z) (-2.6523186143876067+1.8853470951513935j) >>> np.pi**2/6 - np.log(z)*np.log(1 - z) (-2.652318614387606+1.885347095151394j) Plot the function for positive real input. >>> fig, ax = plt.subplots() >>> x = np.linspace(0, 6, 400) >>> ax.plot(x, spence(x)) >>> ax.grid() >>> ax.set_xlabel('x') >>> ax.set_title('spence(x)') >>> plt.show()sph_harm(m, n, theta, phi, out=None) Compute spherical harmonics. The spherical harmonics are defined as .. math:: Y^m_n(\theta,\phi) = \sqrt{\frac{2n+1}{4\pi} \frac{(n-m)!}{(n+m)!}} e^{i m \theta} P^m_n(\cos(\phi)) where :math:`P_n^m` are the associated Legendre functions; see `lpmv`. Parameters ---------- m : array_like Order of the harmonic (int); must have ``|m| <= n``. n : array_like Degree of the harmonic (int); must have ``n >= 0``. This is often denoted by ``l`` (lower case L) in descriptions of spherical harmonics. theta : array_like Azimuthal (longitudinal) coordinate; must be in ``[0, 2*pi]``. phi : array_like Polar (colatitudinal) coordinate; must be in ``[0, pi]``. out : ndarray, optional Optional output array for the function values Returns ------- y_mn : complex scalar or ndarray The harmonic :math:`Y^m_n` sampled at ``theta`` and ``phi``. Notes ----- There are different conventions for the meanings of the input arguments ``theta`` and ``phi``. In SciPy ``theta`` is the azimuthal angle and ``phi`` is the polar angle. It is common to see the opposite convention, that is, ``theta`` as the polar angle and ``phi`` as the azimuthal angle. Note that SciPy's spherical harmonics include the Condon-Shortley phase [2]_ because it is part of `lpmv`. With SciPy's conventions, the first several spherical harmonics are .. math:: Y_0^0(\theta, \phi) &= \frac{1}{2} \sqrt{\frac{1}{\pi}} \\ Y_1^{-1}(\theta, \phi) &= \frac{1}{2} \sqrt{\frac{3}{2\pi}} e^{-i\theta} \sin(\phi) \\ Y_1^0(\theta, \phi) &= \frac{1}{2} \sqrt{\frac{3}{\pi}} \cos(\phi) \\ Y_1^1(\theta, \phi) &= -\frac{1}{2} \sqrt{\frac{3}{2\pi}} e^{i\theta} \sin(\phi). References ---------- .. [1] Digital Library of Mathematical Functions, 14.30. https://dlmf.nist.gov/14.30 .. [2] https://en.wikipedia.org/wiki/Spherical_harmonics#Condon.E2.80.93Shortley_phasestdtrstdtr(df, t, out=None) Student t distribution cumulative distribution function Returns the integral: .. math:: \frac{\Gamma((df+1)/2)}{\sqrt{\pi df} \Gamma(df/2)} \int_{-\infty}^t (1+x^2/df)^{-(df+1)/2}\, dx Parameters ---------- df : array_like Degrees of freedom t : array_like Upper bound of the integral out : ndarray, optional Optional output array for the function results Returns ------- scalar or ndarray Value of the Student t CDF at t See Also -------- stdtridf : inverse of stdtr with respect to `df` stdtrit : inverse of stdtr with respect to `t` scipy.stats.t : student t distribution Notes ----- The student t distribution is also available as `scipy.stats.t`. Calling `stdtr` directly can improve performance compared to the ``cdf`` method of `scipy.stats.t` (see last example below). Examples -------- Calculate the function for ``df=3`` at ``t=1``. >>> import numpy as np >>> from scipy.special import stdtr >>> import matplotlib.pyplot as plt >>> stdtr(3, 1) 0.8044988905221148 Plot the function for three different degrees of freedom. >>> x = np.linspace(-10, 10, 1000) >>> fig, ax = plt.subplots() >>> parameters = [(1, "solid"), (3, "dashed"), (10, "dotted")] >>> for (df, linestyle) in parameters: ... ax.plot(x, stdtr(df, x), ls=linestyle, label=f"$df={df}$") >>> ax.legend() >>> ax.set_title("Student t distribution cumulative distribution function") >>> plt.show() The function can be computed for several degrees of freedom at the same time by providing a NumPy array or list for `df`: >>> stdtr([1, 2, 3], 1) array([0.75 , 0.78867513, 0.80449889]) It is possible to calculate the function at several points for several different degrees of freedom simultaneously by providing arrays for `df` and `t` with shapes compatible for broadcasting. Compute `stdtr` at 4 points for 3 degrees of freedom resulting in an array of shape 3x4. >>> dfs = np.array([[1], [2], [3]]) >>> t = np.array([2, 4, 6, 8]) >>> dfs.shape, t.shape ((3, 1), (4,)) >>> stdtr(dfs, t) array([[0.85241638, 0.92202087, 0.94743154, 0.96041658], [0.90824829, 0.97140452, 0.98666426, 0.99236596], [0.93033702, 0.98599577, 0.99536364, 0.99796171]]) The t distribution is also available as `scipy.stats.t`. Calling `stdtr` directly can be much faster than calling the ``cdf`` method of `scipy.stats.t`. To get the same results, one must use the following parametrization: ``scipy.stats.t(df).cdf(x) = stdtr(df, x)``. >>> from scipy.stats import t >>> df, x = 3, 1 >>> stdtr_result = stdtr(df, x) # this can be faster than below >>> stats_result = t(df).cdf(x) >>> stats_result == stdtr_result # test that results are equal Truestdtridfstdtridf(p, t, out=None) Inverse of `stdtr` vs df Returns the argument df such that stdtr(df, t) is equal to `p`. Parameters ---------- p : array_like Probability t : array_like Upper bound of the integral out : ndarray, optional Optional output array for the function results Returns ------- df : scalar or ndarray Value of `df` such that ``stdtr(df, t) == p`` See Also -------- stdtr : Student t CDF stdtrit : inverse of stdtr with respect to `t` scipy.stats.t : Student t distribution Examples -------- Compute the student t cumulative distribution function for one parameter set. >>> from scipy.special import stdtr, stdtridf >>> df, x = 5, 2 >>> cdf_value = stdtr(df, x) >>> cdf_value 0.9490302605850709 Verify that `stdtridf` recovers the original value for `df` given the CDF value and `x`. >>> stdtridf(cdf_value, x) 5.0stdtritstdtrit(df, p, out=None) The `p`-th quantile of the student t distribution. This function is the inverse of the student t distribution cumulative distribution function (CDF), returning `t` such that `stdtr(df, t) = p`. Returns the argument `t` such that stdtr(df, t) is equal to `p`. Parameters ---------- df : array_like Degrees of freedom p : array_like Probability out : ndarray, optional Optional output array for the function results Returns ------- t : scalar or ndarray Value of `t` such that ``stdtr(df, t) == p`` See Also -------- stdtr : Student t CDF stdtridf : inverse of stdtr with respect to `df` scipy.stats.t : Student t distribution Notes ----- The student t distribution is also available as `scipy.stats.t`. Calling `stdtrit` directly can improve performance compared to the ``ppf`` method of `scipy.stats.t` (see last example below). Examples -------- `stdtrit` represents the inverse of the student t distribution CDF which is available as `stdtr`. Here, we calculate the CDF for ``df`` at ``x=1``. `stdtrit` then returns ``1`` up to floating point errors given the same value for `df` and the computed CDF value. >>> import numpy as np >>> from scipy.special import stdtr, stdtrit >>> import matplotlib.pyplot as plt >>> df = 3 >>> x = 1 >>> cdf_value = stdtr(df, x) >>> stdtrit(df, cdf_value) 0.9999999994418539 Plot the function for three different degrees of freedom. >>> x = np.linspace(0, 1, 1000) >>> parameters = [(1, "solid"), (2, "dashed"), (5, "dotted")] >>> fig, ax = plt.subplots() >>> for (df, linestyle) in parameters: ... ax.plot(x, stdtrit(df, x), ls=linestyle, label=f"$df={df}$") >>> ax.legend() >>> ax.set_ylim(-10, 10) >>> ax.set_title("Student t distribution quantile function") >>> plt.show() The function can be computed for several degrees of freedom at the same time by providing a NumPy array or list for `df`: >>> stdtrit([1, 2, 3], 0.7) array([0.72654253, 0.6172134 , 0.58438973]) It is possible to calculate the function at several points for several different degrees of freedom simultaneously by providing arrays for `df` and `p` with shapes compatible for broadcasting. Compute `stdtrit` at 4 points for 3 degrees of freedom resulting in an array of shape 3x4. >>> dfs = np.array([[1], [2], [3]]) >>> p = np.array([0.2, 0.4, 0.7, 0.8]) >>> dfs.shape, p.shape ((3, 1), (4,)) >>> stdtrit(dfs, p) array([[-1.37638192, -0.3249197 , 0.72654253, 1.37638192], [-1.06066017, -0.28867513, 0.6172134 , 1.06066017], [-0.97847231, -0.27667066, 0.58438973, 0.97847231]]) The t distribution is also available as `scipy.stats.t`. Calling `stdtrit` directly can be much faster than calling the ``ppf`` method of `scipy.stats.t`. To get the same results, one must use the following parametrization: ``scipy.stats.t(df).ppf(x) = stdtrit(df, x)``. >>> from scipy.stats import t >>> df, x = 3, 0.5 >>> stdtrit_result = stdtrit(df, x) # this can be faster than below >>> stats_result = t(df).ppf(x) >>> stats_result == stdtrit_result # test that results are equal Truestruvestruve(v, x, out=None) Struve function. Return the value of the Struve function of order `v` at `x`. The Struve function is defined as, .. math:: H_v(x) = (z/2)^{v + 1} \sum_{n=0}^\infty \frac{(-1)^n (z/2)^{2n}}{\Gamma(n + \frac{3}{2}) \Gamma(n + v + \frac{3}{2})}, where :math:`\Gamma` is the gamma function. Parameters ---------- v : array_like Order of the Struve function (float). x : array_like Argument of the Struve function (float; must be positive unless `v` is an integer). out : ndarray, optional Optional output array for the function results Returns ------- H : scalar or ndarray Value of the Struve function of order `v` at `x`. Notes ----- Three methods discussed in [1]_ are used to evaluate the Struve function: - power series - expansion in Bessel functions (if :math:`|z| < |v| + 20`) - asymptotic large-z expansion (if :math:`z \geq 0.7v + 12`) Rounding errors are estimated based on the largest terms in the sums, and the result associated with the smallest error is returned. See also -------- modstruve: Modified Struve function References ---------- .. [1] NIST Digital Library of Mathematical Functions https://dlmf.nist.gov/11 Examples -------- Calculate the Struve function of order 1 at 2. >>> import numpy as np >>> from scipy.special import struve >>> import matplotlib.pyplot as plt >>> struve(1, 2.) 0.6467637282835622 Calculate the Struve function at 2 for orders 1, 2 and 3 by providing a list for the order parameter `v`. >>> struve([1, 2, 3], 2.) array([0.64676373, 0.28031806, 0.08363767]) Calculate the Struve function of order 1 for several points by providing an array for `x`. >>> points = np.array([2., 5., 8.]) >>> struve(1, points) array([0.64676373, 0.80781195, 0.48811605]) Compute the Struve function for several orders at several points by providing arrays for `v` and `z`. The arrays have to be broadcastable to the correct shapes. >>> orders = np.array([[1], [2], [3]]) >>> points.shape, orders.shape ((3,), (3, 1)) >>> struve(orders, points) array([[0.64676373, 0.80781195, 0.48811605], [0.28031806, 1.56937455, 1.51769363], [0.08363767, 1.50872065, 2.98697513]]) Plot the Struve functions of order 0 to 3 from -10 to 10. >>> fig, ax = plt.subplots() >>> x = np.linspace(-10., 10., 1000) >>> for i in range(4): ... ax.plot(x, struve(i, x), label=f'$H_{i!r}$') >>> ax.legend(ncol=2) >>> ax.set_xlim(-10, 10) >>> ax.set_title(r"Struve functions $H_{\nu}$") >>> plt.show()tandgtandg(x, out=None) Tangent of angle `x` given in degrees. Parameters ---------- x : array_like Angle, given in degrees. out : ndarray, optional Optional output array for the function results. Returns ------- scalar or ndarray Tangent at the input. See Also -------- sindg, cosdg, cotdg Examples -------- >>> import numpy as np >>> import scipy.special as sc It is more accurate than using tangent directly. >>> x = 180 * np.arange(3) >>> sc.tandg(x) array([0., 0., 0.]) >>> np.tan(x * np.pi / 180) array([ 0.0000000e+00, -1.2246468e-16, -2.4492936e-16])tklmbdatklmbda(x, lmbda, out=None) Cumulative distribution function of the Tukey lambda distribution. Parameters ---------- x, lmbda : array_like Parameters out : ndarray, optional Optional output array for the function results Returns ------- cdf : scalar or ndarray Value of the Tukey lambda CDF See Also -------- scipy.stats.tukeylambda : Tukey lambda distribution Examples -------- >>> import numpy as np >>> import matplotlib.pyplot as plt >>> from scipy.special import tklmbda, expit Compute the cumulative distribution function (CDF) of the Tukey lambda distribution at several ``x`` values for `lmbda` = -1.5. >>> x = np.linspace(-2, 2, 9) >>> x array([-2. , -1.5, -1. , -0.5, 0. , 0.5, 1. , 1.5, 2. ]) >>> tklmbda(x, -1.5) array([0.34688734, 0.3786554 , 0.41528805, 0.45629737, 0.5 , 0.54370263, 0.58471195, 0.6213446 , 0.65311266]) When `lmbda` is 0, the function is the logistic sigmoid function, which is implemented in `scipy.special` as `expit`. >>> tklmbda(x, 0) array([0.11920292, 0.18242552, 0.26894142, 0.37754067, 0.5 , 0.62245933, 0.73105858, 0.81757448, 0.88079708]) >>> expit(x) array([0.11920292, 0.18242552, 0.26894142, 0.37754067, 0.5 , 0.62245933, 0.73105858, 0.81757448, 0.88079708]) When `lmbda` is 1, the Tukey lambda distribution is uniform on the interval [-1, 1], so the CDF increases linearly. >>> t = np.linspace(-1, 1, 9) >>> tklmbda(t, 1) array([0. , 0.125, 0.25 , 0.375, 0.5 , 0.625, 0.75 , 0.875, 1. ]) In the following, we generate plots for several values of `lmbda`. The first figure shows graphs for `lmbda` <= 0. >>> styles = ['-', '-.', '--', ':'] >>> fig, ax = plt.subplots() >>> x = np.linspace(-12, 12, 500) >>> for k, lmbda in enumerate([-1.0, -0.5, 0.0]): ... y = tklmbda(x, lmbda) ... ax.plot(x, y, styles[k], label=f'$\lambda$ = {lmbda:-4.1f}') >>> ax.set_title('tklmbda(x, $\lambda$)') >>> ax.set_label('x') >>> ax.legend(framealpha=1, shadow=True) >>> ax.grid(True) The second figure shows graphs for `lmbda` > 0. The dots in the graphs show the bounds of the support of the distribution. >>> fig, ax = plt.subplots() >>> x = np.linspace(-4.2, 4.2, 500) >>> lmbdas = [0.25, 0.5, 1.0, 1.5] >>> for k, lmbda in enumerate(lmbdas): ... y = tklmbda(x, lmbda) ... ax.plot(x, y, styles[k], label=f'$\lambda$ = {lmbda}') >>> ax.set_prop_cycle(None) >>> for lmbda in lmbdas: ... ax.plot([-1/lmbda, 1/lmbda], [0, 1], '.', ms=8) >>> ax.set_title('tklmbda(x, $\lambda$)') >>> ax.set_xlabel('x') >>> ax.legend(framealpha=1, shadow=True) >>> ax.grid(True) >>> plt.tight_layout() >>> plt.show() The CDF of the Tukey lambda distribution is also implemented as the ``cdf`` method of `scipy.stats.tukeylambda`. In the following, ``tukeylambda.cdf(x, -0.5)`` and ``tklmbda(x, -0.5)`` compute the same values: >>> from scipy.stats import tukeylambda >>> x = np.linspace(-2, 2, 9) >>> tukeylambda.cdf(x, -0.5) array([0.21995157, 0.27093858, 0.33541677, 0.41328161, 0.5 , 0.58671839, 0.66458323, 0.72906142, 0.78004843]) >>> tklmbda(x, -0.5) array([0.21995157, 0.27093858, 0.33541677, 0.41328161, 0.5 , 0.58671839, 0.66458323, 0.72906142, 0.78004843]) The implementation in ``tukeylambda`` also provides location and scale parameters, and other methods such as ``pdf()`` (the probability density function) and ``ppf()`` (the inverse of the CDF), so for working with the Tukey lambda distribution, ``tukeylambda`` is more generally useful. The primary advantage of ``tklmbda`` is that it is significantly faster than ``tukeylambda.cdf``.voigt_profilevoigt_profile(x, sigma, gamma, out=None) Voigt profile. The Voigt profile is a convolution of a 1-D Normal distribution with standard deviation ``sigma`` and a 1-D Cauchy distribution with half-width at half-maximum ``gamma``. If ``sigma = 0``, PDF of Cauchy distribution is returned. Conversely, if ``gamma = 0``, PDF of Normal distribution is returned. If ``sigma = gamma = 0``, the return value is ``Inf`` for ``x = 0``, and ``0`` for all other ``x``. Parameters ---------- x : array_like Real argument sigma : array_like The standard deviation of the Normal distribution part gamma : array_like The half-width at half-maximum of the Cauchy distribution part out : ndarray, optional Optional output array for the function values Returns ------- scalar or ndarray The Voigt profile at the given arguments Notes ----- It can be expressed in terms of Faddeeva function .. math:: V(x; \sigma, \gamma) = \frac{Re[w(z)]}{\sigma\sqrt{2\pi}}, .. math:: z = \frac{x + i\gamma}{\sqrt{2}\sigma} where :math:`w(z)` is the Faddeeva function. See Also -------- wofz : Faddeeva function References ---------- .. [1] https://en.wikipedia.org/wiki/Voigt_profile Examples -------- Calculate the function at point 2 for ``sigma=1`` and ``gamma=1``. >>> from scipy.special import voigt_profile >>> import numpy as np >>> import matplotlib.pyplot as plt >>> voigt_profile(2, 1., 1.) 0.09071519942627544 Calculate the function at several points by providing a NumPy array for `x`. >>> values = np.array([-2., 0., 5]) >>> voigt_profile(values, 1., 1.) array([0.0907152 , 0.20870928, 0.01388492]) Plot the function for different parameter sets. >>> fig, ax = plt.subplots(figsize=(8, 8)) >>> x = np.linspace(-10, 10, 500) >>> parameters_list = [(1.5, 0., "solid"), (1.3, 0.5, "dashed"), ... (0., 1.8, "dotted"), (1., 1., "dashdot")] >>> for params in parameters_list: ... sigma, gamma, linestyle = params ... voigt = voigt_profile(x, sigma, gamma) ... ax.plot(x, voigt, label=rf"$\sigma={sigma},\, \gamma={gamma}$", ... ls=linestyle) >>> ax.legend() >>> plt.show() Verify visually that the Voigt profile indeed arises as the convolution of a normal and a Cauchy distribution. >>> from scipy.signal import convolve >>> x, dx = np.linspace(-10, 10, 500, retstep=True) >>> def gaussian(x, sigma): ... return np.exp(-0.5 * x**2/sigma**2)/(sigma * np.sqrt(2*np.pi)) >>> def cauchy(x, gamma): ... return gamma/(np.pi * (np.square(x)+gamma**2)) >>> sigma = 2 >>> gamma = 1 >>> gauss_profile = gaussian(x, sigma) >>> cauchy_profile = cauchy(x, gamma) >>> convolved = dx * convolve(cauchy_profile, gauss_profile, mode="same") >>> voigt = voigt_profile(x, sigma, gamma) >>> fig, ax = plt.subplots(figsize=(8, 8)) >>> ax.plot(x, gauss_profile, label="Gauss: $G$", c='b') >>> ax.plot(x, cauchy_profile, label="Cauchy: $C$", c='y', ls="dashed") >>> xx = 0.5*(x[1:] + x[:-1]) # midpoints >>> ax.plot(xx, convolved[1:], label="Convolution: $G * C$", ls='dashdot', ... c='k') >>> ax.plot(x, voigt, label="Voigt", ls='dotted', c='r') >>> ax.legend() >>> plt.show()wofzwofz(z, out=None) Faddeeva function Returns the value of the Faddeeva function for complex argument:: exp(-z**2) * erfc(-i*z) Parameters ---------- z : array_like complex argument out : ndarray, optional Optional output array for the function results Returns ------- scalar or ndarray Value of the Faddeeva function See Also -------- dawsn, erf, erfc, erfcx, erfi References ---------- .. [1] Steven G. Johnson, Faddeeva W function implementation. http://ab-initio.mit.edu/Faddeeva Examples -------- >>> import numpy as np >>> from scipy import special >>> import matplotlib.pyplot as plt >>> x = np.linspace(-3, 3) >>> z = special.wofz(x) >>> plt.plot(x, z.real, label='wofz(x).real') >>> plt.plot(x, z.imag, label='wofz(x).imag') >>> plt.xlabel('$x$') >>> plt.legend(framealpha=1, shadow=True) >>> plt.grid(alpha=0.25) >>> plt.show()wright_bessel(a, b, x, out=None) Wright's generalized Bessel function. Wright's generalized Bessel function is an entire function and defined as .. math:: \Phi(a, b; x) = \sum_{k=0}^\infty \frac{x^k}{k! \Gamma(a k + b)} See also [1]. Parameters ---------- a : array_like of float a >= 0 b : array_like of float b >= 0 x : array_like of float x >= 0 out : ndarray, optional Optional output array for the function results Returns ------- scalar or ndarray Value of the Wright's generalized Bessel function Notes ----- Due to the compexity of the function with its three parameters, only non-negative arguments are implemented. Examples -------- >>> from scipy.special import wright_bessel >>> a, b, x = 1.5, 1.1, 2.5 >>> wright_bessel(a, b-1, x) 4.5314465939443025 Now, let us verify the relation .. math:: \Phi(a, b-1; x) = a x \Phi(a, b+a; x) + (b-1) \Phi(a, b; x) >>> a * x * wright_bessel(a, b+a, x) + (b-1) * wright_bessel(a, b, x) 4.5314465939443025 References ---------- .. [1] Digital Library of Mathematical Functions, 10.46. https://dlmf.nist.gov/10.46.E1wrightomega(z, out=None) Wright Omega function. Defined as the solution to .. math:: \omega + \log(\omega) = z where :math:`\log` is the principal branch of the complex logarithm. Parameters ---------- z : array_like Points at which to evaluate the Wright Omega function out : ndarray, optional Optional output array for the function values Returns ------- omega : scalar or ndarray Values of the Wright Omega function Notes ----- .. versionadded:: 0.19.0 The function can also be defined as .. math:: \omega(z) = W_{K(z)}(e^z) where :math:`K(z) = \lceil (\Im(z) - \pi)/(2\pi) \rceil` is the unwinding number and :math:`W` is the Lambert W function. The implementation here is taken from [1]_. See Also -------- lambertw : The Lambert W function References ---------- .. [1] Lawrence, Corless, and Jeffrey, "Algorithm 917: Complex Double-Precision Evaluation of the Wright :math:`\omega` Function." ACM Transactions on Mathematical Software, 2012. :doi:`10.1145/2168773.2168779`. Examples -------- >>> import numpy as np >>> from scipy.special import wrightomega, lambertw >>> wrightomega([-2, -1, 0, 1, 2]) array([0.12002824, 0.27846454, 0.56714329, 1. , 1.5571456 ]) Complex input: >>> wrightomega(3 + 5j) (1.5804428632097158+3.8213626783287937j) Verify that ``wrightomega(z)`` satisfies ``w + log(w) = z``: >>> w = -5 + 4j >>> wrightomega(w + np.log(w)) (-5+4j) Verify the connection to ``lambertw``: >>> z = 0.5 + 3j >>> wrightomega(z) (0.0966015889280649+1.4937828458191993j) >>> lambertw(np.exp(z)) (0.09660158892806493+1.4937828458191993j) >>> z = 0.5 + 4j >>> wrightomega(z) (-0.3362123489037213+2.282986001579032j) >>> lambertw(np.exp(z), k=1) (-0.33621234890372115+2.282986001579032j)xlog1pyxlog1py(x, y, out=None) Compute ``x*log1p(y)`` so that the result is 0 if ``x = 0``. Parameters ---------- x : array_like Multiplier y : array_like Argument out : ndarray, optional Optional output array for the function results Returns ------- z : scalar or ndarray Computed x*log1p(y) Notes ----- .. versionadded:: 0.13.0 Examples -------- This example shows how the function can be used to calculate the log of the probability mass function for a geometric discrete random variable. The probability mass function of the geometric distribution is defined as follows: .. math:: f(k) = (1-p)^{k-1} p where :math:`p` is the probability of a single success and :math:`1-p` is the probability of a single failure and :math:`k` is the number of trials to get the first success. >>> import numpy as np >>> from scipy.special import xlog1py >>> p = 0.5 >>> k = 100 >>> _pmf = np.power(1 - p, k - 1) * p >>> _pmf 7.888609052210118e-31 If we take k as a relatively large number the value of the probability mass function can become very low. In such cases taking the log of the pmf would be more suitable as the log function can change the values to a scale that is more appropriate to work with. >>> _log_pmf = xlog1py(k - 1, -p) + np.log(p) >>> _log_pmf -69.31471805599453 We can confirm that we get a value close to the original pmf value by taking the exponential of the log pmf. >>> _orig_pmf = np.exp(_log_pmf) >>> np.isclose(_pmf, _orig_pmf) Truexlogyxlogy(x, y, out=None) Compute ``x*log(y)`` so that the result is 0 if ``x = 0``. Parameters ---------- x : array_like Multiplier y : array_like Argument out : ndarray, optional Optional output array for the function results Returns ------- z : scalar or ndarray Computed x*log(y) Notes ----- The log function used in the computation is the natural log. .. versionadded:: 0.13.0 Examples -------- We can use this function to calculate the binary logistic loss also known as the binary cross entropy. This loss function is used for binary classification problems and is defined as: .. math:: L = 1/n * \sum_{i=0}^n -(y_i*log(y\_pred_i) + (1-y_i)*log(1-y\_pred_i)) We can define the parameters `x` and `y` as y and y_pred respectively. y is the array of the actual labels which over here can be either 0 or 1. y_pred is the array of the predicted probabilities with respect to the positive class (1). >>> import numpy as np >>> from scipy.special import xlogy >>> y = np.array([0, 1, 0, 1, 1, 0]) >>> y_pred = np.array([0.3, 0.8, 0.4, 0.7, 0.9, 0.2]) >>> n = len(y) >>> loss = -(xlogy(y, y_pred) + xlogy(1 - y, 1 - y_pred)).sum() >>> loss /= n >>> loss 0.29597052165495025 A lower loss is usually better as it indicates that the predictions are similar to the actual labels. In this example since our predicted probabilties are close to the actual labels, we get an overall loss that is reasonably low and appropriate.y0y0(x, out=None) Bessel function of the second kind of order 0. Parameters ---------- x : array_like Argument (float). out : ndarray, optional Optional output array for the function results Returns ------- Y : scalar or ndarray Value of the Bessel function of the second kind of order 0 at `x`. Notes ----- The domain is divided into the intervals [0, 5] and (5, infinity). In the first interval a rational approximation :math:`R(x)` is employed to compute, .. math:: Y_0(x) = R(x) + \frac{2 \log(x) J_0(x)}{\pi}, where :math:`J_0` is the Bessel function of the first kind of order 0. In the second interval, the Hankel asymptotic expansion is employed with two rational functions of degree 6/6 and 7/7. This function is a wrapper for the Cephes [1]_ routine `y0`. See also -------- j0: Bessel function of the first kind of order 0 yv: Bessel function of the first kind References ---------- .. [1] Cephes Mathematical Functions Library, http://www.netlib.org/cephes/ Examples -------- Calculate the function at one point: >>> from scipy.special import y0 >>> y0(1.) 0.08825696421567697 Calculate at several points: >>> import numpy as np >>> y0(np.array([0.5, 2., 3.])) array([-0.44451873, 0.51037567, 0.37685001]) Plot the function from 0 to 10. >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots() >>> x = np.linspace(0., 10., 1000) >>> y = y0(x) >>> ax.plot(x, y) >>> plt.show()y1y1(x, out=None) Bessel function of the second kind of order 1. Parameters ---------- x : array_like Argument (float). out : ndarray, optional Optional output array for the function results Returns ------- Y : scalar or ndarray Value of the Bessel function of the second kind of order 1 at `x`. Notes ----- The domain is divided into the intervals [0, 8] and (8, infinity). In the first interval a 25 term Chebyshev expansion is used, and computing :math:`J_1` (the Bessel function of the first kind) is required. In the second, the asymptotic trigonometric representation is employed using two rational functions of degree 5/5. This function is a wrapper for the Cephes [1]_ routine `y1`. See also -------- j1: Bessel function of the first kind of order 1 yn: Bessel function of the second kind yv: Bessel function of the second kind References ---------- .. [1] Cephes Mathematical Functions Library, http://www.netlib.org/cephes/ Examples -------- Calculate the function at one point: >>> from scipy.special import y1 >>> y1(1.) -0.7812128213002888 Calculate at several points: >>> import numpy as np >>> y1(np.array([0.5, 2., 3.])) array([-1.47147239, -0.10703243, 0.32467442]) Plot the function from 0 to 10. >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots() >>> x = np.linspace(0., 10., 1000) >>> y = y1(x) >>> ax.plot(x, y) >>> plt.show()ynyn(n, x, out=None) Bessel function of the second kind of integer order and real argument. Parameters ---------- n : array_like Order (integer). x : array_like Argument (float). out : ndarray, optional Optional output array for the function results Returns ------- Y : scalar or ndarray Value of the Bessel function, :math:`Y_n(x)`. Notes ----- Wrapper for the Cephes [1]_ routine `yn`. The function is evaluated by forward recurrence on `n`, starting with values computed by the Cephes routines `y0` and `y1`. If `n = 0` or 1, the routine for `y0` or `y1` is called directly. See also -------- yv : For real order and real or complex argument. y0: faster implementation of this function for order 0 y1: faster implementation of this function for order 1 References ---------- .. [1] Cephes Mathematical Functions Library, http://www.netlib.org/cephes/ Examples -------- Evaluate the function of order 0 at one point. >>> from scipy.special import yn >>> yn(0, 1.) 0.08825696421567697 Evaluate the function at one point for different orders. >>> yn(0, 1.), yn(1, 1.), yn(2, 1.) (0.08825696421567697, -0.7812128213002888, -1.6506826068162546) The evaluation for different orders can be carried out in one call by providing a list or NumPy array as argument for the `v` parameter: >>> yn([0, 1, 2], 1.) array([ 0.08825696, -0.78121282, -1.65068261]) Evaluate the function at several points for order 0 by providing an array for `z`. >>> import numpy as np >>> points = np.array([0.5, 3., 8.]) >>> yn(0, points) array([-0.44451873, 0.37685001, 0.22352149]) If `z` is an array, the order parameter `v` must be broadcastable to the correct shape if different orders shall be computed in one call. To calculate the orders 0 and 1 for an 1D array: >>> orders = np.array([[0], [1]]) >>> orders.shape (2, 1) >>> yn(orders, points) array([[-0.44451873, 0.37685001, 0.22352149], [-1.47147239, 0.32467442, -0.15806046]]) Plot the functions of order 0 to 3 from 0 to 10. >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots() >>> x = np.linspace(0., 10., 1000) >>> for i in range(4): ... ax.plot(x, yn(i, x), label=f'$Y_{i!r}$') >>> ax.set_ylim(-3, 1) >>> ax.legend() >>> plt.show()yvyv(v, z, out=None) Bessel function of the second kind of real order and complex argument. Parameters ---------- v : array_like Order (float). z : array_like Argument (float or complex). out : ndarray, optional Optional output array for the function results Returns ------- Y : scalar or ndarray Value of the Bessel function of the second kind, :math:`Y_v(x)`. Notes ----- For positive `v` values, the computation is carried out using the AMOS [1]_ `zbesy` routine, which exploits the connection to the Hankel Bessel functions :math:`H_v^{(1)}` and :math:`H_v^{(2)}`, .. math:: Y_v(z) = \frac{1}{2\imath} (H_v^{(1)} - H_v^{(2)}). For negative `v` values the formula, .. math:: Y_{-v}(z) = Y_v(z) \cos(\pi v) + J_v(z) \sin(\pi v) is used, where :math:`J_v(z)` is the Bessel function of the first kind, computed using the AMOS routine `zbesj`. Note that the second term is exactly zero for integer `v`; to improve accuracy the second term is explicitly omitted for `v` values such that `v = floor(v)`. See also -------- yve : :math:`Y_v` with leading exponential behavior stripped off. y0: faster implementation of this function for order 0 y1: faster implementation of this function for order 1 References ---------- .. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions of a Complex Argument and Nonnegative Order", http://netlib.org/amos/ Examples -------- Evaluate the function of order 0 at one point. >>> from scipy.special import yv >>> yv(0, 1.) 0.088256964215677 Evaluate the function at one point for different orders. >>> yv(0, 1.), yv(1, 1.), yv(1.5, 1.) (0.088256964215677, -0.7812128213002889, -1.102495575160179) The evaluation for different orders can be carried out in one call by providing a list or NumPy array as argument for the `v` parameter: >>> yv([0, 1, 1.5], 1.) array([ 0.08825696, -0.78121282, -1.10249558]) Evaluate the function at several points for order 0 by providing an array for `z`. >>> import numpy as np >>> points = np.array([0.5, 3., 8.]) >>> yv(0, points) array([-0.44451873, 0.37685001, 0.22352149]) If `z` is an array, the order parameter `v` must be broadcastable to the correct shape if different orders shall be computed in one call. To calculate the orders 0 and 1 for an 1D array: >>> orders = np.array([[0], [1]]) >>> orders.shape (2, 1) >>> yv(orders, points) array([[-0.44451873, 0.37685001, 0.22352149], [-1.47147239, 0.32467442, -0.15806046]]) Plot the functions of order 0 to 3 from 0 to 10. >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots() >>> x = np.linspace(0., 10., 1000) >>> for i in range(4): ... ax.plot(x, yv(i, x), label=f'$Y_{i!r}$') >>> ax.set_ylim(-3, 1) >>> ax.legend() >>> plt.show()yveyve(v, z, out=None) Exponentially scaled Bessel function of the second kind of real order. Returns the exponentially scaled Bessel function of the second kind of real order `v` at complex `z`:: yve(v, z) = yv(v, z) * exp(-abs(z.imag)) Parameters ---------- v : array_like Order (float). z : array_like Argument (float or complex). out : ndarray, optional Optional output array for the function results Returns ------- Y : scalar or ndarray Value of the exponentially scaled Bessel function. See Also -------- yv: Unscaled Bessel function of the second kind of real order. Notes ----- For positive `v` values, the computation is carried out using the AMOS [1]_ `zbesy` routine, which exploits the connection to the Hankel Bessel functions :math:`H_v^{(1)}` and :math:`H_v^{(2)}`, .. math:: Y_v(z) = \frac{1}{2\imath} (H_v^{(1)} - H_v^{(2)}). For negative `v` values the formula, .. math:: Y_{-v}(z) = Y_v(z) \cos(\pi v) + J_v(z) \sin(\pi v) is used, where :math:`J_v(z)` is the Bessel function of the first kind, computed using the AMOS routine `zbesj`. Note that the second term is exactly zero for integer `v`; to improve accuracy the second term is explicitly omitted for `v` values such that `v = floor(v)`. Exponentially scaled Bessel functions are useful for large `z`: for these, the unscaled Bessel functions can easily under-or overflow. References ---------- .. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions of a Complex Argument and Nonnegative Order", http://netlib.org/amos/ Examples -------- Compare the output of `yv` and `yve` for large complex arguments for `z` by computing their values for order ``v=1`` at ``z=1000j``. We see that `yv` returns nan but `yve` returns a finite number: >>> import numpy as np >>> from scipy.special import yv, yve >>> v = 1 >>> z = 1000j >>> yv(v, z), yve(v, z) ((nan+nanj), (-0.012610930256928629+7.721967686709076e-19j)) For real arguments for `z`, `yve` returns the same as `yv` up to floating point errors. >>> v, z = 1, 1000 >>> yv(v, z), yve(v, z) (-0.02478433129235178, -0.02478433129235179) The function can be evaluated for several orders at the same time by providing a list or NumPy array for `v`: >>> yve([1, 2, 3], 1j) array([-0.20791042+0.14096627j, 0.38053618-0.04993878j, 0.00815531-1.66311097j]) In the same way, the function can be evaluated at several points in one call by providing a list or NumPy array for `z`: >>> yve(1, np.array([1j, 2j, 3j])) array([-0.20791042+0.14096627j, -0.21526929+0.01205044j, -0.19682671+0.00127278j]) It is also possible to evaluate several orders at several points at the same time by providing arrays for `v` and `z` with broadcasting compatible shapes. Compute `yve` for two different orders `v` and three points `z` resulting in a 2x3 array. >>> v = np.array([[1], [2]]) >>> z = np.array([3j, 4j, 5j]) >>> v.shape, z.shape ((2, 1), (3,)) >>> yve(v, z) array([[-1.96826713e-01+1.27277544e-03j, -1.78750840e-01+1.45558819e-04j, -1.63972267e-01+1.73494110e-05j], [1.94960056e-03-1.11782545e-01j, 2.02902325e-04-1.17626501e-01j, 2.27727687e-05-1.17951906e-01j]])zetaczetac(x, out=None) Riemann zeta function minus 1. This function is defined as .. math:: \zeta(x) = \sum_{k=2}^{\infty} 1 / k^x, where ``x > 1``. For ``x < 1`` the analytic continuation is computed. For more information on the Riemann zeta function, see [dlmf]_. Parameters ---------- x : array_like of float Values at which to compute zeta(x) - 1 (must be real). out : ndarray, optional Optional output array for the function results Returns ------- scalar or ndarray Values of zeta(x) - 1. See Also -------- zeta Examples -------- >>> import numpy as np >>> from scipy.special import zetac, zeta Some special values: >>> zetac(2), np.pi**2/6 - 1 (0.64493406684822641, 0.6449340668482264) >>> zetac(-1), -1.0/12 - 1 (-1.0833333333333333, -1.0833333333333333) Compare ``zetac(x)`` to ``zeta(x) - 1`` for large `x`: >>> zetac(60), zeta(60) - 1 (8.673617380119933e-19, 0.0) References ---------- .. [dlmf] NIST Digital Library of Mathematical Functions https://dlmf.nist.gov/25init scipy.special._ufuncs__enter__geterr_ufuncs__name__ must be set to a string object__qualname__ must be set to a string objectfunction's dictionary may not be deletedsetting function's dictionary to a non-dict__defaults__ must be set to a tuple object__kwdefaults__ must be set to a dict object__annotations__ must be set to a dict object%.200s() takes no arguments (%zd given)%.200s() takes exactly one argument (%zd given)Bad call flags in __Pyx_CyFunction_Call. METH_OLDARGS is no longer supported!%.200s() takes no keyword argumentsunbound method %.200S() needs an argument__int__ returned non-int (type %.200s). The ability to return an instance of a strict subclass of int is deprecated, and may be removed in a future version of Python.__%.4s__ returned non-%.4s (type %.200s)'%.200s' object is not subscriptablecannot fit '%.200s' into an index-sized integer%s() got multiple values for keyword argument '%U'%s() got an unexpected keyword argument '%U'_cython_0_29_37Shared Cython type %.200s is not a type objectShared Cython type %.200s has the wrong size, try recompilingscipy/special/_ufuncs.cpython-312-aarch64-linux-gnu.so.p/_ufuncs.c%s (%s:%d)can't convert negative value to sf_error_tvalue too large to convert to sf_error_tintan integer is requiredcan't convert negative value to sf_action_tvalue too large to convert to sf_action_t?scipy.special/%s: (%s) %sscipy.special/%s: %sscipy.specialSpecialFunctionWarningSpecialFunctionErrorairy:airye:ive:ive(kv):iv:iv(kv):jve:jve(yve):jv:jv(yv):yv:yv(jv):yve:kv:kve:hankel1:hankel1e:hankel2:hankel2e:(Fortran) input parameter %d is out of rangeAnswer appears to be lower than lowest search bound (%g)Answer appears to be higher than highest search bound (%g)Two parameters that should sum to 1.0 do notComputational errorUnknown errorchyp2f1cexp1cexpiklvnacem_cvamcm1msm1mcm2msm2pmvmemory allocation errorprolate_segvoblate_segvprolate_aswfa_nocvoblate_aswfa_nocvprolate_aswfaoblate_aswfaprolate_radial1_nocvprolate_radial2_nocvprolate_radial1prolate_radial2oblate_radial1_nocvoblate_radial2_nocvoblate_radial1oblate_radial2incbetlbetaellpeellpjellpkellikfloating point division by zerofloating point underflowfloating point overflowfloating point invalid valuefunc_doc__doc__func_name__name____qualname____self__func_dict__dict__func_globals__globals__func_closure__closure__func_code__code__func_defaults__defaults____kwdefaults____annotations____module____reduce__no errorsingularityunderflowoverflowtoo slow convergenceloss of precisionno result obtaineddomain errorother errorcython_function_or_methodGammaigamJvikv_asymptotic_uniformiv(iv_asymptotic)ikv_temme(CF1_ik)ikv_temme(temme_ik_series)ikv_temme(CF2_ik)incbizeta'NoneType' object is not iterableneed more than %zd value%.1s to unpacktoo many values to unpack (expected %zd)'%.50s' object has no attribute '%U'calling %R should have returned an instance of BaseException, not %Rraise: exception class must be a subclass of BaseException%.200s does not export expected C variable %.200svoid *C variable %.200s.%.200s has wrong signature (expected %.500s, got %.500s)co_argcountco_posonlyargcountco_kwonlyargcountco_nlocalsco_stacksizeco_flagsco_codeco_constsco_namesco_varnamesco_freevarsco_cellvarsco_linetablereplace%.200s.%.200s is not a type object%.200s.%.200s size changed, may indicate binary incompatibility. Expected %zd from C header, got %zd from PyObject%s.%s size changed, may indicate binary incompatibility. 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Parameters ---------- all : {'ignore', 'warn' 'raise'}, optional Set treatment for all type of special-function errors at once. The options are: - 'ignore' Take no action when the error occurs - 'warn' Print a `SpecialFunctionWarning` when the error occurs (via the Python `warnings` module) - 'raise' Raise a `SpecialFunctionError` when the error occurs. The default is to not change the current behavior. If behaviors for additional categories of special-function errors are specified, then ``all`` is applied first, followed by the additional categories. singular : {'ignore', 'warn', 'raise'}, optional Treatment for singularities. underflow : {'ignore', 'warn', 'raise'}, optional Treatment for underflow. overflow : {'ignore', 'warn', 'raise'}, optional Treatment for overflow. slow : {'ignore', 'warn', 'raise'}, optional Treatment for slow convergence. loss : {'ignore', 'warn', 'raise'}, optional Treatment for loss of accuracy. no_result : {'ignore', 'warn', 'raise'}, optional Treatment for failing to find a result. domain : {'ignore', 'warn', 'raise'}, optional Treatment for an invalid argument to a function. arg : {'ignore', 'warn', 'raise'}, optional Treatment for an invalid parameter to a function. other : {'ignore', 'warn', 'raise'}, optional Treatment for an unknown error. Returns ------- olderr : dict Dictionary containing the old settings. See Also -------- geterr : get the current way of handling special-function errors errstate : context manager for special-function error handling numpy.seterr : similar numpy function for floating-point errors Examples -------- >>> import scipy.special as sc >>> from pytest import raises >>> sc.gammaln(0) inf >>> olderr = sc.seterr(singular='raise') >>> with raises(sc.SpecialFunctionError): ... sc.gammaln(0) ... >>> _ = sc.seterr(**olderr) We can also raise for every category except one. >>> olderr = sc.seterr(all='raise', singular='ignore') >>> sc.gammaln(0) inf >>> with raises(sc.SpecialFunctionError): ... sc.spence(-1) ... >>> _ = sc.seterr(**olderr) Get the current way of handling special-function errors. Returns ------- err : dict A dictionary with keys "singular", "underflow", "overflow", "slow", "loss", "no_result", "domain", "arg", and "other", whose values are from the strings "ignore", "warn", and "raise". The keys represent possible special-function errors, and the values define how these errors are handled. See Also -------- seterr : set how special-function errors are handled errstate : context manager for special-function error handling numpy.geterr : similar numpy function for floating-point errors Notes ----- For complete documentation of the types of special-function errors and treatment options, see `seterr`. 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