MZdddlmZddlmZddlmZddlmZddlm Z ddl m Z m Z m Z ddlmZmZddlmZmZmZdd lmZdd lmZmZdd lmZdd lmZdd lmZm Z m!Z!m"Z"ddl#m$Z$ddl%m&Z&m'Z'ddl(m)Z)m*Z*m+Z+ddl,m-Z-m.Z.m/Z/m0Z0m1Z1ddl2m3Z3m4Z4m5Z5ddl6m7Z7ddl8m9Z9ddl:m;Z;mGdde=Z?Gdde=Z@Gdde=ZAGd d!e=ZBGd"d#e=ZCd$ZDGd%d&e=ZEd'ZFd(ZGGd)d*eEZHGd+d,eEZIGd-d.eEZJGd/d0eJZKGd1d2eJZLdAd3ZMGd4d5e ZNGd6d7eNZOGd8d9eNZPGd:d;eNZQGd<d=eNZRGd>d?e ZSy@)Bwraps)S)Add)cacheit)Expr)FunctionArgumentIndexError_mexpand)fuzzy_or fuzzy_not)RationalpiI)Pow)DummyWild)sympify) factorial)sincoscsccot)ceiling)explog)cbrtsqrtroot)Absreim polar_lift unpolarify)gammadigamma uppergamma)hyper)spherical_bessel_fn)mpworkprecc`eZdZdZedZedZedZd dZ dZ dZ dZ d Z y ) BesselBasea Abstract base class for Bessel-type functions. This class is meant to reduce code duplication. All Bessel-type functions can 1) be differentiated, with the derivatives expressed in terms of similar functions, and 2) be rewritten in terms of other Bessel-type functions. Here, Bessel-type functions are assumed to have one complex parameter. To use this base class, define class attributes ``_a`` and ``_b`` such that ``2*F_n' = -_a*F_{n+1} + b*F_{n-1}``. c |jdS)z( The order of the Bessel-type function. rargsselfs @/usr/lib/python3/dist-packages/sympy/functions/special/bessel.pyorderzBesselBase.order4yy|c |jdS)z+ The argument of the Bessel-type function. r/r1s r3argumentzBesselBase.argument9r5r6cyNclsnuzs r3evalzBesselBase.eval>sr6c |dk7r t|||jdz |j|jdz |jz|j dz |j|jdz|jzz SNr8)r _b __class__r4r9_ar2argindexs r3fdiffzBesselBase.fdiffBsp q=$T84 4 DNN4::>4==II DNN4::>4==IIJ Kr6c|j}|jdur8|j|jj |j SyNF)r9is_extended_negativerFr4 conjugater2r@s r3_eval_conjugatezBesselBase._eval_conjugateHsB MM ! !U *>>$**"6"6"8!++-H H +r6c |j|j}}|j|ry|j||sy|j ||}|j rKt |ttttttfs |jst|jStt!|j|jgSrL)r4r9has_eval_is_meromorphicsubs is_integer isinstancebesseljbesselihn1hn2jnynis_zeror is_infiniter )r2xar?r@z0s r3rSzBesselBase._eval_is_meromorphicMs DMMA 66!9%%a+ VVAq\ ==$'3R DERZZ 002::r~~">?@@r6c D|j|j|j}}}|jr|dz jri|j |j z||dz |jzd|j z|dz z||dz |jz|z zS|dzjrhd|j z|dzz||dz|jz|z |j |j z||dz|jzz S|SNr8rD) r4r9rFis_real is_positiverGrE_eval_expand_func is_negative)r2hintsr?r@fs r3rfzBesselBase._eval_expand_funcZs ::t}}dnnqA ::Q##(261)G)G)II$'' 26*1R!VQ<+I+I+KKAMNOq&%%$'' 26*1R!VQ<+I+I+KKAM"q&! (F(F(HHIJ r6c ddlm}||S)Nr) besselsimp)sympy.simplify.simplifyrk)r2kwargsrks r3_eval_simplifyzBesselBase._eval_simplifyes6$r6NrD)__name__ __module__ __qualname____doc__propertyr4r9 classmethodrArJrPrSrfrnr<r6r3r-r-$s_ K I A  r6r-ceZdZdZej Zej ZedZ dZ dZ dZ d fd Z dZd fd ZxZS) rWa4 Bessel function of the first kind. Explanation =========== The Bessel $J$ function of order $\nu$ is defined to be the function satisfying Bessel's differential equation .. math :: z^2 \frac{\mathrm{d}^2 w}{\mathrm{d}z^2} + z \frac{\mathrm{d}w}{\mathrm{d}z} + (z^2 - \nu^2) w = 0, with Laurent expansion .. math :: J_\nu(z) = z^\nu \left(\frac{1}{\Gamma(\nu + 1) 2^\nu} + O(z^2) \right), if $\nu$ is not a negative integer. If $\nu=-n \in \mathbb{Z}_{<0}$ *is* a negative integer, then the definition is .. math :: J_{-n}(z) = (-1)^n J_n(z). Examples ======== Create a Bessel function object: >>> from sympy import besselj, jn >>> from sympy.abc import z, n >>> b = besselj(n, z) Differentiate it: >>> b.diff(z) besselj(n - 1, z)/2 - besselj(n + 1, z)/2 Rewrite in terms of spherical Bessel functions: >>> b.rewrite(jn) sqrt(2)*sqrt(z)*jn(n - 1/2, z)/sqrt(pi) Access the parameter and argument: >>> b.order n >>> b.argument z See Also ======== bessely, besseli, besselk References ========== .. [1] Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 9", Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables .. [2] Luke, Y. L. (1969), The Special Functions and Their Approximations, Volume 1 .. [3] https://en.wikipedia.org/wiki/Bessel_function .. [4] https://functions.wolfram.com/Bessel-TypeFunctions/BesselJ/ c|jr|jrtjS|jr|jdust |j rtj St |jr|jdurtjS|jrtjS|tjtjfvrtj S|jr||z| | zzt|| zS|jr_|jr"tj| zt| |zS|j!t"}|rt"|zt%||zS|jrt'|}||k7rPt||S|j)\}}|dk7r,t+d|zt,z|zt"zt||zSt'|}||k7r t||Sy)NFTrrD)r]rOnerUr!reZerorgComplexInfinity is_imaginaryNaNInfinityNegativeInfinitycould_extract_minus_signrW NegativeOneextract_multiplicativelyrrXr$extract_branch_factorrrr>r?r@newznnnus r3rAz besselj.evals 99zzuu --BJJ%$7BrF0II I "r6c |td|ztz t|tjz |j zSr)rrr[rHalfr9rs r3_eval_rewrite_as_jnzbesselj._eval_rewrite_as_jns,AaCF|BrAFF{DMM:::r6c|j\}} |j|}|j|\}}|jr||zd|zt |dzzz S|j r\|dk(rdn|}|||zz} | j s=tdt|td|zdzzdz z ztt|zz S|Stt|3|||S#t$r|cYSwxYw)NrDr8r) r0as_leading_termNotImplementedErroras_coeff_exponentrer%rgrrrsuperrW_eval_as_leading_term r2r_logxcdirr?r@argcesignrFs r3rzbesselj._eval_as_leading_terms A ##A&C$$Q'1 ==7ArE%Q-/0 0 ]] 1tDT1W9D##Aws1r1R4!8}Q#677RT BBKWd9!T4HH# K sC C('C(cV|j\}}|jr|jryyyNTr0rUis_extended_realr2r?r@s r3_eval_is_extended_realzbesselj._eval_is_extended_real( A ==Q//0=r6cddlm}|j\}} |j|\}} | j rt|| z } |||z|} |dz j||||j} | tjur| St| dz| zj} | |zt|dzz }|g}td| dzdzD]>}|| |||zzz z}t|| zj}|j|@t!|| zSt"t$|#||||S#tt f$r|cYSwxYwNrOrderrDr8)sympy.series.orderrr0leadterm ValueErrorrrer _eval_nseriesremoveOrryr r%rangeappendrrrWr2r_rrrrr?r@_rnewnorttermskrFs r3rzbesselj._eval_nseriessW - A ZZ]FAs ??1S5>DadAA1##Aq$5==?AAFF{!Q$!#,,.Ab5rAv&DA1tax!m, ArAvJ' *335 7Q; Wd1!QdCC'/0 K sD55E E Nrr)rprqrrrsrrxrGrErurArrrrrr __classcell__rFs@r3rWrWjsXBH B B!#!#F<J;I* DDr6rWceZdZdZej Zej ZedZ dZ dZ dZ d fd Z dZd fd ZxZS) ra` Bessel function of the second kind. Explanation =========== The Bessel $Y$ function of order $\nu$ is defined as .. math :: Y_\nu(z) = \lim_{\mu \to \nu} \frac{J_\mu(z) \cos(\pi \mu) - J_{-\mu}(z)}{\sin(\pi \mu)}, where $J_\mu(z)$ is the Bessel function of the first kind. It is a solution to Bessel's equation, and linearly independent from $J_\nu$. Examples ======== >>> from sympy import bessely, yn >>> from sympy.abc import z, n >>> b = bessely(n, z) >>> b.diff(z) bessely(n - 1, z)/2 - bessely(n + 1, z)/2 >>> b.rewrite(yn) sqrt(2)*sqrt(z)*yn(n - 1/2, z)/sqrt(pi) See Also ======== besselj, besseli, besselk References ========== .. [1] https://functions.wolfram.com/Bessel-TypeFunctions/BesselY/ c|jrh|jrtjSt|jdurtjSt|jrtj S|tj tjfvrtjS|ttj zk(r0tttz|dzzdz tj zS|ttjzk(r1tt tz|dzzdz tj zS|jr3|jr"tj| zt| |zSyy)NFr8rD)r]rr~r!rzr|r}ryrrrrUrrrr=s r3rAz bessely.evalFs 99zz)))B5((((Buu Q//0 066M !** qtR!V}Q'!**4 4 !$$$ $r"ub1f~a'(1::5 5 ==**,}}s+GRCO;;- r6c |jdur@tt|ztt|zt ||zt | |z zSyrL)rUrrrrWrs r3_eval_rewrite_as_besseljz bessely._eval_rewrite_as_besseljZsF ==E !r"u:s2b5z'"a.87B3?JK K "r6c d|j|j}|r|jtSyr;)rr0rewriterXr2r?r@rmajs r3rz bessely._eval_rewrite_as_besseli^/ *T * *DII 6 ::g& & r6c |td|ztz t|tjz |j zSr)rrr\rrr9rs r3_eval_rewrite_as_ynzbessely._eval_rewrite_as_yncs,AaCF|baffdmm<<@=M=M1 }YrAv%66r9STSYSYHQ3)R " %56Q!,,8VWJ(J78HHQUHVCJ ]] 1tDT1W9D##AwRU1Wq[2a4%7!8 81SBq1rRStAS=T;TVWXYVY;Z Z[\`abcdad\eefjkmfnnnKWd9!T4HH'# K sF88 GGcV|j\}}|jr|jryyyrr0rUrers r3rzbessely._eval_is_extended_real& A ==Q]]+=r6cBddlm}|j\}} |j|\}} | j r;|jr.t|| z } t||} dtz t|dz z| zj||||} gg}} |||z|}|dz j||||j}|tjur|St!|dz|zj}|tjkDr|| zt#|dz ztz }| j%|t'd|D][}||z |z}|tjk(r |||z z}n|||z z}t!||zj}| j%|]||ztt#|zz }|t)|dztj*z z}|j%|t'd| dzdzD]a}|| |||zzz z}t!||zj}|t)||zdzt)|dzzz}|j%|c| t-| z t-|z St.t0|3||||S#tt f$r|cYSwxYwr)rrr0rrrrerUrrWrrrrrryr rrrr&rrrrr2r_rrrrr?r@rrrbnr`brrrrrrdenomprFs r3rzbessely._eval_nseriess - A ZZ]FAs ??r}}1S5>DQBB$AaC#221atDArqAadAA1##Aq$5==?AAFF{!Q$!#,,.AAFF{B3x "q& 11"4q"#A!VQJE! %$TNQ.779DHHTN#2r)B-'(Agb1fo 45D HHTN1tax!m, aRAF_$a[1_--/'!b&1*-A>?   sAw;a( (Wd1!QdCCK/0 K sJ JJrr)rprqrrrsrrxrGrErurArrrrrrrrs@r3rrsV&P B B<<&L' =I2 /D/Dr6rceZdZdZej Zej ZedZ dZ dZ dZ dZ d fd Zd fd ZxZS) rXa Modified Bessel function of the first kind. Explanation =========== The Bessel $I$ function is a solution to the modified Bessel equation .. math :: z^2 \frac{\mathrm{d}^2 w}{\mathrm{d}z^2} + z \frac{\mathrm{d}w}{\mathrm{d}z} + (z^2 + \nu^2)^2 w = 0. It can be defined as .. math :: I_\nu(z) = i^{-\nu} J_\nu(iz), where $J_\nu(z)$ is the Bessel function of the first kind. Examples ======== >>> from sympy import besseli >>> from sympy.abc import z, n >>> besseli(n, z).diff(z) besseli(n - 1, z)/2 + besseli(n + 1, z)/2 See Also ======== besselj, bessely, besselk References ========== .. [1] https://functions.wolfram.com/Bessel-TypeFunctions/BesselI/ c|jr|jrtjS|jr|jdust |j rtj St |jr|jdurtjS|jrtjSt|tjtjfvrtj S|tjurtjS|tjurd|ztjzS|jr||z| | zzt|| zS|jrL|jr t| |S|j!t"}|rt"| zt%|| zS|jrt'|}||k7rPt||S|j)\}}|dk7r,t+d|zt,z|zt"zt||zSt'|}||k7r t||Sy)NFTrrD)r]rrxrUr!reryrgrzr{r|r"r}r~rrXrrrWr$rrrrs r3rAz besseli.evals 99zzuu --BJJ%$7BrF> C  C cddlm}|j\}} |j|\}} | j rt|| z } |||z|} |dz j||||j} | tjur| St| dz| zj} | |zt|dzz }|g}td| dzdzD]=}|| |||zzz z}t|| zj}|j|?t!|| zSt"t$|#||||S#tt f$r|cYSwxYwr)rrr0rrrrerrrrryr r%rrrrrXrs r3rzbesseli._eval_nseries.sU - A ZZ]FAs ??1S5>DadAA1##Aq$5==?AAFF{!Q$!#,,.Ab5rAv&DA1tax!m, 1b1f:& *335 7Q; Wd1!QdCC'/0 K sD44EErr)rprqrrrsrrxrGrErurArrrrrrrrs@r3rXrXsY%N %%B B%#%#N<' E I*DDr6rXceZdZdZej Zej ZedZ dZ dZ dZ dZ dZd fd Zd fd ZxZS) besselka Modified Bessel function of the second kind. Explanation =========== The Bessel $K$ function of order $\nu$ is defined as .. math :: K_\nu(z) = \lim_{\mu \to \nu} \frac{\pi}{2} \frac{I_{-\mu}(z) -I_\mu(z)}{\sin(\pi \mu)}, where $I_\mu(z)$ is the modified Bessel function of the first kind. It is a solution of the modified Bessel equation, and linearly independent from $Y_\nu$. Examples ======== >>> from sympy import besselk >>> from sympy.abc import z, n >>> besselk(n, z).diff(z) -besselk(n - 1, z)/2 - besselk(n + 1, z)/2 See Also ======== besselj, besseli, bessely References ========== .. [1] https://functions.wolfram.com/Bessel-TypeFunctions/BesselK/ c|jrh|jrtjSt|jdurtjSt|jrtj S|tjt tjzt tjzfvrtjS|jr|jr t| |SyyrL) r]rr}r!rzr|rr~ryrUrrr=s r3rAz besselk.evalws 99zzzz!B5((((Buu Qqzz\1Q-?-?+?@ @66M ==**,sA&- r6c |jdur7ttt|zzt| |t||z zdz Sy)NFrD)rUrrrXrs r3rz besselk._eval_rewrite_as_besselisB ==E !c"R%j='2#q/GBN"BCAE E "r6c d|j|j}|r|jtSyr;)rr0rrW)r2r?r@rmais r3rz besselk._eval_rewrite_as_besseljrr6c d|j|j}|r|jtSyr;rrs r3rz besselk._eval_rewrite_as_besselyrr6c d|j|j}|r|jtSyr;)rr0rr\)r2r?r@rmays r3rzbesselk._eval_rewrite_as_yns. *T * *DII 6 ::b> ! r6cV|j\}}|jr|jryyyrrrs r3rzbesselk._eval_is_extended_realrr6c|j\}} |j|}|j|\}}|jrd|dz zt |dz zt ||z} |jr|dz | zt|dz zdz ntj} d|z|dz |zzdt|zz t|dztjz z} t| | | gj||}|S|jr+ttt!| ztd|zz St"t$|O|||S#t$r|cYSwxYw)Nrr8rDr)r0rrrrerrXrrryr&rrrgrrrrrr) r2r_rrr?r@rrrrrrrFs r3rzbesselk._eval_as_leading_termsT A ##A&C$$Q'1 ==r1u c!A#h.wr1~=H<>;K;K!s|Ib1f$55a7QRQWQWHr1Q3)+Qy}_=wrAvQRQ]Q]?]^J(J78HHQUHVCJ ]]8CG#D1I- -Wd9!T4HH# K sE EEc:ddlm}|j\}} |j|\}} | j r7|jr*t|| z } t||} d|dz zt|dz z| zj||||} gg}} |||z|}|dz j||||j}|tjur|St|dz|zj}|tjkDr|| zt!|dz zdz }| j#|t%d|D][}||z |z}|tjk(r |||z z}n|||z z}t||zj}| j#|]||zd|zzdt!|zz }|t'|dztj(z z}|j#|t%d| dzdzD]`}|||||zzz z}t||zj}|t'||zdzt'|dzzz}|j#|b| t+| zt+|zSt,t.|/||||S#tt f$r|cYSwxYw)Nrrrr8rD)rrr0rrrrerUrrXrrrrryr rrrr&rrrrrs r3rzbesselk._eval_nseriess - A ZZ]FAs ??r}}1S5>DQBQAaC(+::1atLArqAadAA1##Aq$5==?AAFF{!Q$!#,,.AAFF{B3x "q& 11!3q"#AVQJE! %$TNQ.779DHHTN#2rBh)B-0Agb1fo 45D HHTN1tax!m, Q1r6 ^#a[1_--/'!b&1*-A>?   sAw;a( (Wd1!QdCCK/0 K sJJJrr)rprqrrrsrrxrGrErurArrrrrrrrrs@r3rrNs]#J B %%B ' 'F' ' "  I*/D/Dr6rcFeZdZdZej Zej ZdZy)hankel1a Hankel function of the first kind. Explanation =========== This function is defined as .. math :: H_\nu^{(1)} = J_\nu(z) + iY_\nu(z), where $J_\nu(z)$ is the Bessel function of the first kind, and $Y_\nu(z)$ is the Bessel function of the second kind. It is a solution to Bessel's equation. Examples ======== >>> from sympy import hankel1 >>> from sympy.abc import z, n >>> hankel1(n, z).diff(z) hankel1(n - 1, z)/2 - hankel1(n + 1, z)/2 See Also ======== hankel2, besselj, bessely References ========== .. [1] https://functions.wolfram.com/Bessel-TypeFunctions/HankelH1/ c|j}|jdur2t|jj |j SyrL)r9rMhankel2r4rNrOs r3rPzhankel1._eval_conjugate> MM ! !U *4:://11;;=A A +r6N rprqrrrsrrxrGrErPr<r6r3rrs""H B BBr6rcFeZdZdZej Zej ZdZy)ra Hankel function of the second kind. Explanation =========== This function is defined as .. math :: H_\nu^{(2)} = J_\nu(z) - iY_\nu(z), where $J_\nu(z)$ is the Bessel function of the first kind, and $Y_\nu(z)$ is the Bessel function of the second kind. It is a solution to Bessel's equation, and linearly independent from $H_\nu^{(1)}$. Examples ======== >>> from sympy import hankel2 >>> from sympy.abc import z, n >>> hankel2(n, z).diff(z) hankel2(n - 1, z)/2 - hankel2(n + 1, z)/2 See Also ======== hankel1, besselj, bessely References ========== .. [1] https://functions.wolfram.com/Bessel-TypeFunctions/HankelH2/ c|j}|jdur2t|jj |j SyrL)r9rMrr4rNrOs r3rPzhankel2._eval_conjugate=rr6Nrr<r6r3rrs"#J B BBr6rc.tfd}|S)Nc2|jr |||Syr;)rU)r2r?r@fns r3gzassume_integer_order..gDs ==dB? " r6r)rrs` r3assume_integer_orderrCs  2Y## Hr6c$eZdZdZdZdZddZy)SphericalBesselBasea- Base class for spherical Bessel functions. These are thin wrappers around ordinary Bessel functions, since spherical Bessel functions differ from the ordinary ones just by a slight change in order. To use this class, define the ``_eval_evalf()`` and ``_expand()`` methods. c td)z@ Expand self into a polynomial. Nu is guaranteed to be Integer. expansionrr2rhs r3_expandzSphericalBesselBase._expandWs !+..r6c V|jjr|jdi|S|SNr<)r4 is_Integerrrs r3rfz%SphericalBesselBase._eval_expand_func[s( :: 4<<(%( ( r6c|dk7r t|||j|jdz |j||jdzz|jz z SrC)r rFr4r9rHs r3rJzSphericalBesselBase.fdiff`sS q=$T84 4~~djj1ndmm< DJJN #DMM 12 2r6Nro)rprqrrrsrrfrJr<r6r3rrKs / 2r6rct||t|ztj|dzzt| dz |zt |zzSNr8)r)rrrrrr@s r3_jnr gsM 1 %c!f , MMAE "#6rAvq#A A#a& H IJr6ctj|dzzt| dz |zt|zt||t |zz Sr )rrr)rrr s r3_ynrlsK MMAE "%8!a%C CCF J 1 %c!f , -.r6c>eZdZdZedZdZdZdZdZ dZ y) r[a Spherical Bessel function of the first kind. Explanation =========== This function is a solution to the spherical Bessel equation .. math :: z^2 \frac{\mathrm{d}^2 w}{\mathrm{d}z^2} + 2z \frac{\mathrm{d}w}{\mathrm{d}z} + (z^2 - \nu(\nu + 1)) w = 0. It can be defined as .. math :: j_\nu(z) = \sqrt{\frac{\pi}{2z}} J_{\nu + \frac{1}{2}}(z), where $J_\nu(z)$ is the Bessel function of the first kind. The spherical Bessel functions of integral order are calculated using the formula: .. math:: j_n(z) = f_n(z) \sin{z} + (-1)^{n+1} f_{-n-1}(z) \cos{z}, where the coefficients $f_n(z)$ are available as :func:`sympy.polys.orthopolys.spherical_bessel_fn`. Examples ======== >>> from sympy import Symbol, jn, sin, cos, expand_func, besselj, bessely >>> z = Symbol("z") >>> nu = Symbol("nu", integer=True) >>> print(expand_func(jn(0, z))) sin(z)/z >>> expand_func(jn(1, z)) == sin(z)/z**2 - cos(z)/z True >>> expand_func(jn(3, z)) (-6/z**2 + 15/z**4)*sin(z) + (1/z - 15/z**3)*cos(z) >>> jn(nu, z).rewrite(besselj) sqrt(2)*sqrt(pi)*sqrt(1/z)*besselj(nu + 1/2, z)/2 >>> jn(nu, z).rewrite(bessely) (-1)**nu*sqrt(2)*sqrt(pi)*sqrt(1/z)*bessely(-nu - 1/2, z)/2 >>> jn(2, 5.2+0.3j).evalf(20) 0.099419756723640344491 - 0.054525080242173562897*I See Also ======== besselj, bessely, besselk, yn References ========== .. [1] https://dlmf.nist.gov/10.47 c(|jrT|jrtjS|jr,|jrtj Stj S|tjtjfvrtj Syr;) r]rrxrUreryrzr~r}r=s r3rAzjn.evalsa 99zzuu >>66M,,, ##QZZ0 066M 1r6c httd|zz t|tjz|zSr)rrrWrrrs r3rzjn._eval_rewrite_as_besseljs(B!H~QVV Q 777r6c tj|zttd|zz zt | tj z |zSr)rrrrrrrs r3rzjn._eval_rewrite_as_besselys8}}b 4AaC>1GRC!&&L!4LLLr6c Jtj|zt| dz |zSr )rrr\rs r3rzjn._eval_rewrite_as_yns"}}r"Ra^33r6c Bt|j|jSr;)r r4r9rs r3rz jn._expand4::t}}--r6cx|jjr$|jtj |Syr;r4rrrW _eval_evalfr2precs r3rzjn._eval_evalf. :: <<(44T: : !r6N) rprqrrrsrurArrrrrr<r6r3r[r[rs68r  8M4.;r6r[cBeZdZdZedZedZdZdZdZ y)r\a Spherical Bessel function of the second kind. Explanation =========== This function is another solution to the spherical Bessel equation, and linearly independent from $j_n$. It can be defined as .. math :: y_\nu(z) = \sqrt{\frac{\pi}{2z}} Y_{\nu + \frac{1}{2}}(z), where $Y_\nu(z)$ is the Bessel function of the second kind. For integral orders $n$, $y_n$ is calculated using the formula: .. math:: y_n(z) = (-1)^{n+1} j_{-n-1}(z) Examples ======== >>> from sympy import Symbol, yn, sin, cos, expand_func, besselj, bessely >>> z = Symbol("z") >>> nu = Symbol("nu", integer=True) >>> print(expand_func(yn(0, z))) -cos(z)/z >>> expand_func(yn(1, z)) == -cos(z)/z**2-sin(z)/z True >>> yn(nu, z).rewrite(besselj) (-1)**(nu + 1)*sqrt(2)*sqrt(pi)*sqrt(1/z)*besselj(-nu - 1/2, z)/2 >>> yn(nu, z).rewrite(bessely) sqrt(2)*sqrt(pi)*sqrt(1/z)*bessely(nu + 1/2, z)/2 >>> yn(2, 5.2+0.3j).evalf(20) 0.18525034196069722536 + 0.014895573969924817587*I See Also ======== besselj, bessely, besselk, jn References ========== .. [1] https://dlmf.nist.gov/10.47 c tj|dzzttd|zz zt | tj z |zSrc)rrrrrWrrs r3rzyn._eval_rewrite_as_besseljs<}}r!t$tB!H~5aff a8PPPr6c httd|zz t|tjz|zSr)rrrrrrs r3rzyn._eval_rewrite_as_besselys(B!H~QVV Q 777r6c Ptj|dzzt| dz |zSr )rrr[rs r3rzyn._eval_rewrite_as_jns&}}rAv&RC!GQ77r6c Bt|j|jSr;)rr4r9rs r3rz yn._expandrr6cx|jjr$|jtj |Syr;)r4rrrrrs r3rzyn._eval_evalfrr6N) rprqrrrsrrrrrrr<r6r3r\r\sA-\QQ888.;r6r\cJeZdZedZedZdZdZdZdZ dZ y) SphericalHankelBasec |j}ttd|zz t|tj z||t ztj|dzzzt| tj z |zzzSrC)_hankel_kind_signrrrWrrrrr2r?r@rmhkss r3rz,SphericalHankelBase._eval_rewrite_as_besseljsq $$B!H~wrAFF{A6"1uQ]]RT%::7B3 >r6c |j}t||jt|tzt||zzSr;)r%r[rr\rr&s r3rz'SphericalHankelBase._eval_rewrite_as_yn s7$$"ay  $s1uRAY66r6c |j}t|||tzt||j tzzSr;)r%r[rr\rr&s r3rz'SphericalHankelBase._eval_rewrite_as_jn$s8$$"ay3q5B!2!22!6666r6c |jjr|jdi|S|j}|j}|j}t |||t zt||zzSr)r4rrr9r%r[rr\)r2rhr?r@r's r3rfz%SphericalHankelBase._eval_expand_func(s_ :: 4<<(%( (B A((Cb!9s1uRAY. .r6c |j}|j}|j}t|||tzt ||zzj Sr;)r4r9r%r rrexpand)r2rhrr@r's r3rzSphericalHankelBase._expand1sI JJ MM$$Aq CE#a)O+3355r6cx|jjr$|jtj |Syr;rrs r3rzSphericalHankelBase._eval_evalf@rr6N) rprqrrrrrrrrfrrr<r6r3r#r# sCUU>>77/ 6;r6r#c8eZdZdZej ZedZy)rYa Spherical Hankel function of the first kind. Explanation =========== This function is defined as .. math:: h_\nu^(1)(z) = j_\nu(z) + i y_\nu(z), where $j_\nu(z)$ and $y_\nu(z)$ are the spherical Bessel function of the first and second kinds. For integral orders $n$, $h_n^(1)$ is calculated using the formula: .. math:: h_n^(1)(z) = j_{n}(z) + i (-1)^{n+1} j_{-n-1}(z) Examples ======== >>> from sympy import Symbol, hn1, hankel1, expand_func, yn, jn >>> z = Symbol("z") >>> nu = Symbol("nu", integer=True) >>> print(expand_func(hn1(nu, z))) jn(nu, z) + I*yn(nu, z) >>> print(expand_func(hn1(0, z))) sin(z)/z - I*cos(z)/z >>> print(expand_func(hn1(1, z))) -I*sin(z)/z - cos(z)/z + sin(z)/z**2 - I*cos(z)/z**2 >>> hn1(nu, z).rewrite(jn) (-1)**(nu + 1)*I*jn(-nu - 1, z) + jn(nu, z) >>> hn1(nu, z).rewrite(yn) (-1)**nu*yn(-nu - 1, z) + I*yn(nu, z) >>> hn1(nu, z).rewrite(hankel1) sqrt(2)*sqrt(pi)*sqrt(1/z)*hankel1(nu, z)/2 See Also ======== hn2, jn, yn, hankel1, hankel2 References ========== .. [1] https://dlmf.nist.gov/10.47 c Fttd|zz t||zSr)rrrrs r3_eval_rewrite_as_hankel1zhn1._eval_rewrite_as_hankel1xB!H~gb!n,,r6N) rprqrrrsrrxr%rr1r<r6r3rYrYEs&.`--r6rYc:eZdZdZej ZedZy)rZa Spherical Hankel function of the second kind. Explanation =========== This function is defined as .. math:: h_\nu^(2)(z) = j_\nu(z) - i y_\nu(z), where $j_\nu(z)$ and $y_\nu(z)$ are the spherical Bessel function of the first and second kinds. For integral orders $n$, $h_n^(2)$ is calculated using the formula: .. math:: h_n^(2)(z) = j_{n} - i (-1)^{n+1} j_{-n-1}(z) Examples ======== >>> from sympy import Symbol, hn2, hankel2, expand_func, jn, yn >>> z = Symbol("z") >>> nu = Symbol("nu", integer=True) >>> print(expand_func(hn2(nu, z))) jn(nu, z) - I*yn(nu, z) >>> print(expand_func(hn2(0, z))) sin(z)/z + I*cos(z)/z >>> print(expand_func(hn2(1, z))) I*sin(z)/z - cos(z)/z + sin(z)/z**2 + I*cos(z)/z**2 >>> hn2(nu, z).rewrite(hankel2) sqrt(2)*sqrt(pi)*sqrt(1/z)*hankel2(nu, z)/2 >>> hn2(nu, z).rewrite(jn) -(-1)**(nu + 1)*I*jn(-nu - 1, z) + jn(nu, z) >>> hn2(nu, z).rewrite(yn) (-1)**nu*yn(-nu - 1, z) - I*yn(nu, z) See Also ======== hn1, jn, yn, hankel1, hankel2 References ========== .. [1] https://dlmf.nist.gov/10.47 c Fttd|zz t||zSr)rrrrs r3_eval_rewrite_as_hankel2zhn2._eval_rewrite_as_hankel2r2r6N) rprqrrrsrrxr%rr5r<r6r3rZrZ}s(.`--r6rZc ddlm}dk(rpddlm}ddlm}||}t d|dzDcgc]C}tj|tdzj|t||Ec}Sdk(rdd l m  dd lmfd } n t%dfd} |z} | | | } | g} t |dz D]} | | | |z} | j'| !| Scc}w#t $rdd lmfd } YewxYw)a Zeros of the spherical Bessel function of the first kind. Explanation =========== This returns an array of zeros of $jn$ up to the $k$-th zero. * method = "sympy": uses `mpmath.besseljzero `_ * method = "scipy": uses the `SciPy's sph_jn `_ and `newton `_ to find all roots, which is faster than computing the zeros using a general numerical solver, but it requires SciPy and only works with low precision floating point numbers. (The function used with method="sympy" is a recent addition to mpmath; before that a general solver was used.) Examples ======== >>> from sympy import jn_zeros >>> jn_zeros(2, 4, dps=5) [5.7635, 9.095, 12.323, 15.515] See Also ======== jn, yn, besselj, besselk, bessely Parameters ========== n : integer order of Bessel function k : integer number of zeros to return r)rsympy) besseljzero) dps_to_precr8g?scipy)newton) spherical_jnc|Sr;r<)r_rr<s r3zjn_zeros..s,q!,r6)sph_jnc"|ddS)Nrrr<)r_rr?s r3r>zjn_zeros..s&A,q/"-r6Unknown method.c:dk(r ||}|Std)Nr:rAr)rir_rmethodr;s r3solverzjn_zeros..solvers+ W !Q>> from sympy import airyai >>> from sympy.abc import z >>> airyai(z) airyai(z) Several special values are known: >>> airyai(0) 3**(1/3)/(3*gamma(2/3)) >>> from sympy import oo >>> airyai(oo) 0 >>> airyai(-oo) 0 The Airy function obeys the mirror symmetry: >>> from sympy import conjugate >>> conjugate(airyai(z)) airyai(conjugate(z)) Differentiation with respect to $z$ is supported: >>> from sympy import diff >>> diff(airyai(z), z) airyaiprime(z) >>> diff(airyai(z), z, 2) z*airyai(z) Series expansion is also supported: >>> from sympy import series >>> series(airyai(z), z, 0, 3) 3**(5/6)*gamma(1/3)/(6*pi) - 3**(1/6)*z*gamma(2/3)/(2*pi) + O(z**3) We can numerically evaluate the Airy function to arbitrary precision on the whole complex plane: >>> airyai(-2).evalf(50) 0.22740742820168557599192443603787379946077222541710 Rewrite $\operatorname{Ai}(z)$ in terms of hypergeometric functions: >>> from sympy import hyper >>> airyai(z).rewrite(hyper) -3**(2/3)*z*hyper((), (4/3,), z**3/9)/(3*gamma(1/3)) + 3**(1/3)*hyper((), (2/3,), z**3/9)/(3*gamma(2/3)) See Also ======== airybi: Airy function of the second kind. airyaiprime: Derivative of the Airy function of the first kind. airybiprime: Derivative of the Airy function of the second kind. References ========== .. [1] https://en.wikipedia.org/wiki/Airy_function .. [2] https://dlmf.nist.gov/9 .. [3] https://encyclopediaofmath.org/wiki/Airy_functions .. [4] https://mathworld.wolfram.com/AiryFunctions.html r8Tc|jr|tjurtjS|tjurtjS|tj urtjS|j r6tjdtddzttddzz S|j r6tjdtddzttddzz Sy)NrrD) is_Numberrr|r}ryr~r]rxrr%r>rs r3rAz airyai.evals ==aee|uu  "vv ***vv uu8Aq> 1E(1a.4I IJJ ;;55Ax1~-hq!n0EEF F r6cT|dk(rt|jdSt||Nr8r) airyaiprimer0r rHs r3rJz airyai.fdiff) q=tyy|, ,$T84 4r6c T|dkrtjSt|}t|dkDr|d}t d|z| zt d|z|dzzzt t |tddztddzzzt|zt|dz tddzzt t |tddztddzzt|dzzt|dz tddzzz |zStjdtddzt zz t|tjztdz zt tddt z|tjzzzt|z t d|z|zzS)Nrr8rrrDr) rryrlenrrrrrr%rxrr_previous_termsrs r3 taylor_termzairyai.taylor_terms q566M A>"Q&"2&aqb)4719A*>>s2qRSUVGWZbcdfgZhGhCi?jjktuvkwwacHQN23458Qx1~=MPXYZ\]P^=^9_5`ajklopkpaq5qrwxyz{x{GHIKLMyMsN6NORSSTq(1a.034uagqt^7LLsS[\]_`SabdSdfghihmhmfmSnOoo!! %(,Q A~67r6c tdd}tdd}t| tdd}t|jr0|t | zt | ||zt |||zzzSyNr8rrDrrr!rgrrWr2r@rmotttr`s r3rzairyai._eval_rewrite_as_besseljsp a^ a^ HQN # a5  dA2h;'2#r!t"4wr2a47H"HI I r6c htdd}tdd}t|tdd}t|jr/|t |zt | ||zt |||zz zS|t||t | ||zz|t|| zt |||zzz zSrrrrr!rerrXrts r3rzairyai._eval_rewrite_as_besselis a^ a^ 8Aq> " a5  d1g:"bd!3gb"Q$6G!GH Hs1bz'2#r!t"44qQ}WRQSTUQUEV7VVW Wr6c >tjdtddzttddzz }|t ddttddzz }|t gtddg|dzdz z|t gtddg|dzdz zz S)NrrDr8 r)rrxrr%rr(r2r@rmpf1pf2s r3_eval_rewrite_as_hyperzairyai._eval_rewrite_as_hyperseeq(1a.(x1~)>>?41:eHQN334U2A/Aa883rHUVXYNK[]^`a]abc]cAd;dddr6c ~|jd}|j}t|dk(r|j}t d|g}t d|g}t d|g}t d|g}|j ||||zz|zz} | | |}d|zj r| |}| |}| |}|||zz|z||z|||zzzz } |||zz|||zzz} tj| tjzt| z| tjz tdz t| zz zSyyy Nrr8r)excludedmrr) r0 free_symbolsrmpoprmatchrUrrrxrdrairybi r2rhrsymbsr@rrrrMpfnewargs r3rfzairyai._eval_expand_funcsRiil   u:? AS1#&AS1#&AS1#&AS1#&A !Qq!tVaK-(A}aDaC##!A!A!Aad(Q!Q$QqS/:BAXAaC0F66b155j&.%@BJPTUVPWCWX^_eXfCf%fgg $  r6Nr8rprqrrrsnargs unbranchedrurArJ staticmethodrrprrr~rfr<r6r3rdrd$sbVp EJ G G5   7  7JXe hr6rdcbeZdZdZdZdZedZd dZe e dZ dZ dZ d Zd Zy ) ra The Airy function $\operatorname{Bi}$ of the second kind. Explanation =========== The Airy function $\operatorname{Bi}(z)$ is defined to be the function satisfying Airy's differential equation .. math:: \frac{\mathrm{d}^2 w(z)}{\mathrm{d}z^2} - z w(z) = 0. Equivalently, for real $z$ .. math:: \operatorname{Bi}(z) := \frac{1}{\pi} \int_0^\infty \exp\left(-\frac{t^3}{3} + z t\right) + \sin\left(\frac{t^3}{3} + z t\right) \mathrm{d}t. Examples ======== Create an Airy function object: >>> from sympy import airybi >>> from sympy.abc import z >>> airybi(z) airybi(z) Several special values are known: >>> airybi(0) 3**(5/6)/(3*gamma(2/3)) >>> from sympy import oo >>> airybi(oo) oo >>> airybi(-oo) 0 The Airy function obeys the mirror symmetry: >>> from sympy import conjugate >>> conjugate(airybi(z)) airybi(conjugate(z)) Differentiation with respect to $z$ is supported: >>> from sympy import diff >>> diff(airybi(z), z) airybiprime(z) >>> diff(airybi(z), z, 2) z*airybi(z) Series expansion is also supported: >>> from sympy import series >>> series(airybi(z), z, 0, 3) 3**(1/3)*gamma(1/3)/(2*pi) + 3**(2/3)*z*gamma(2/3)/(2*pi) + O(z**3) We can numerically evaluate the Airy function to arbitrary precision on the whole complex plane: >>> airybi(-2).evalf(50) -0.41230258795639848808323405461146104203453483447240 Rewrite $\operatorname{Bi}(z)$ in terms of hypergeometric functions: >>> from sympy import hyper >>> airybi(z).rewrite(hyper) 3**(1/6)*z*hyper((), (4/3,), z**3/9)/gamma(1/3) + 3**(5/6)*hyper((), (2/3,), z**3/9)/(3*gamma(2/3)) See Also ======== airyai: Airy function of the first kind. airyaiprime: Derivative of the Airy function of the first kind. airybiprime: Derivative of the Airy function of the second kind. References ========== .. [1] https://en.wikipedia.org/wiki/Airy_function .. [2] https://dlmf.nist.gov/9 .. [3] https://encyclopediaofmath.org/wiki/Airy_functions .. [4] https://mathworld.wolfram.com/AiryFunctions.html r8Tc|jr|tjurtjS|tjurtjS|tjurtj S|j r6tjdtddzttddzz S|j r6tjdtddzttddzz Sy)Nrr8rD) rfrr|r}r~ryr]rxrr%rgs r3rAz airybi.eval.s ==aee|uu  "zz!***vv uu8Aq> 1E(1a.4I IJJ ;;55Ax1~-hq!n0EEF F r6cT|dk(rt|jdSt||ri) airybiprimer0r rHs r3rJz airybi.fdiff=rkr6c l|dkrtjSt|}t|dkDr|d}t d|zt t tddtz|tjzzzt|tjz tdz z|tjzt ttddtz|tjzzzt|dz tdz zz |zStjtddtzz t|tjztdz zt t tddtz|tjzzzt|z t d|z|zzS)Nrr8rrrDr)rryrrmrr rrrrxrrrrr%rns r3rpzairybi.taylor_termCsw q566M A>"Q&"2&Q CHQN2,=q155y,I(J$KKiYZ]^]b]bYbdefgdhXhNiiaee)s3x1~b/@!aff*/M+N'OOR[]^ab]bdefgdh\hRiikmnoptAqz"}-q155y!A$6F0GG#cRZ[\^_R`acRcefijininenRoNpJqq!! %(,Q A~67r6c tdd}tdd}t| tdd}t|jr0t | dz t | ||zt |||zz zSyrrrsrts r3rzairybi._eval_rewrite_as_besseljRsp a^ a^ HQN # a5  1:"bd!3gb"Q$6G!GH H r6c tdd}tdd}t|tdd}t|jr8t |t dz t | ||zt |||zzzSt||}t|| }t ||t | ||zz||zt |||zzzzSrrrxr2r@rmrurvr`rrs r3rzairybi._eval_rewrite_as_besseliYs a^ a^ 8Aq> " a5  747?grc2a4&872r!t;L&LM MAr AAs A8QwsBqD11AaCBqD8I4IIJ Jr6c 8tjtddtt ddzz }|tddztt ddz }|t gt ddg|dzdz z|t gt ddg|dzdz zzS)NrrrDr8rzr)rrxrr%rr(r{s r3r~zairybi._eval_rewrite_as_hyperdseetAqz%A"778Q lU8Aq>22U2A/Aa883rHUVXYNK[]^`a]abc]cAd;dddr6c ~|jd}|j}t|dk(r|j}t d|g}t d|g}t d|g}t d|g}|j ||||zz|zz} | | |}d|zj r| |}| |}| |}|||zz|z||z|||zzzz } |||zz|||zzz} tjtdtj| z zt| ztj| zt| zzzSyyyr) r0rrmrrrrUrrrrxrdrrs r3rfzairybi._eval_expand_funcisPiil   u:? AS1#&AS1#&AS1#&AS1#&A !Qq!tVaK-(A}aDaC##!A!A!Aad(Q!Q$QqS/:BAXAaC0F66T!Waeebj%9&.%HAEETVJX^_eXfKf%fgg $  r6Nrrr<r6r3rrsbXt EJ G G5   7  7I Ke hr6rcNeZdZdZdZdZedZd dZdZ dZ dZ d Z d Z y ) rja% The derivative $\operatorname{Ai}^\prime$ of the Airy function of the first kind. Explanation =========== The Airy function $\operatorname{Ai}^\prime(z)$ is defined to be the function .. math:: \operatorname{Ai}^\prime(z) := \frac{\mathrm{d} \operatorname{Ai}(z)}{\mathrm{d} z}. Examples ======== Create an Airy function object: >>> from sympy import airyaiprime >>> from sympy.abc import z >>> airyaiprime(z) airyaiprime(z) Several special values are known: >>> airyaiprime(0) -3**(2/3)/(3*gamma(1/3)) >>> from sympy import oo >>> airyaiprime(oo) 0 The Airy function obeys the mirror symmetry: >>> from sympy import conjugate >>> conjugate(airyaiprime(z)) airyaiprime(conjugate(z)) Differentiation with respect to $z$ is supported: >>> from sympy import diff >>> diff(airyaiprime(z), z) z*airyai(z) >>> diff(airyaiprime(z), z, 2) z*airyaiprime(z) + airyai(z) Series expansion is also supported: >>> from sympy import series >>> series(airyaiprime(z), z, 0, 3) -3**(2/3)/(3*gamma(1/3)) + 3**(1/3)*z**2/(6*gamma(2/3)) + O(z**3) We can numerically evaluate the Airy function to arbitrary precision on the whole complex plane: >>> airyaiprime(-2).evalf(50) 0.61825902074169104140626429133247528291577794512415 Rewrite $\operatorname{Ai}^\prime(z)$ in terms of hypergeometric functions: >>> from sympy import hyper >>> airyaiprime(z).rewrite(hyper) 3**(1/3)*z**2*hyper((), (5/3,), z**3/9)/(6*gamma(2/3)) - 3**(2/3)*hyper((), (1/3,), z**3/9)/(3*gamma(1/3)) See Also ======== airyai: Airy function of the first kind. airybi: Airy function of the second kind. airybiprime: Derivative of the Airy function of the second kind. References ========== .. [1] https://en.wikipedia.org/wiki/Airy_function .. [2] https://dlmf.nist.gov/9 .. [3] https://encyclopediaofmath.org/wiki/Airy_functions .. 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AS1#&AS1#&AS1#&AS1#&A !Qq!tVaK-(A}aD aC##!A!A!AQ$QqS/a!Q$h]:BAXAaC0F66b155j+f2E%EaeeUYZ[U\H\]hio]pHp%pqq $  r6NrrprqrrrsrrrurArJrrrr~rfr<r6r3rjrjsKOb EJOO5 , B Je rr6rjcNeZdZdZdZdZedZd dZdZ dZ dZ d Z d Z y ) ra6 The derivative $\operatorname{Bi}^\prime$ of the Airy function of the first kind. Explanation =========== The Airy function $\operatorname{Bi}^\prime(z)$ is defined to be the function .. math:: \operatorname{Bi}^\prime(z) := \frac{\mathrm{d} \operatorname{Bi}(z)}{\mathrm{d} z}. Examples ======== Create an Airy function object: >>> from sympy import airybiprime >>> from sympy.abc import z >>> airybiprime(z) airybiprime(z) Several special values are known: >>> airybiprime(0) 3**(1/6)/gamma(1/3) >>> from sympy import oo >>> airybiprime(oo) oo >>> airybiprime(-oo) 0 The Airy function obeys the mirror symmetry: >>> from sympy import conjugate >>> conjugate(airybiprime(z)) airybiprime(conjugate(z)) Differentiation with respect to $z$ is supported: >>> from sympy import diff >>> diff(airybiprime(z), z) z*airybi(z) >>> diff(airybiprime(z), z, 2) z*airybiprime(z) + airybi(z) Series expansion is also supported: >>> from sympy import series >>> series(airybiprime(z), z, 0, 3) 3**(1/6)/gamma(1/3) + 3**(5/6)*z**2/(6*gamma(2/3)) + O(z**3) We can numerically evaluate the Airy function to arbitrary precision on the whole complex plane: >>> airybiprime(-2).evalf(50) 0.27879516692116952268509756941098324140300059345163 Rewrite $\operatorname{Bi}^\prime(z)$ in terms of hypergeometric functions: >>> from sympy import hyper >>> airybiprime(z).rewrite(hyper) 3**(5/6)*z**2*hyper((), (5/3,), z**3/9)/(6*gamma(2/3)) + 3**(1/6)*hyper((), (1/3,), z**3/9)/gamma(1/3) See Also ======== airyai: Airy function of the first kind. airybi: Airy function of the second kind. airyaiprime: Derivative of the Airy function of the first kind. References ========== .. [1] https://en.wikipedia.org/wiki/Airy_function .. [2] https://dlmf.nist.gov/9 .. [3] https://encyclopediaofmath.org/wiki/Airy_functions .. 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Explanation =========== The Marcum Q-function is defined by the meromorphic continuation of .. math:: Q_m(a, b) = a^{- m + 1} \int_{b}^{\infty} x^{m} e^{- \frac{a^{2}}{2} - \frac{x^{2}}{2}} I_{m - 1}\left(a x\right)\, dx Examples ======== >>> from sympy import marcumq >>> from sympy.abc import m, a, b >>> marcumq(m, a, b) marcumq(m, a, b) Special values: >>> marcumq(m, 0, b) uppergamma(m, b**2/2)/gamma(m) >>> marcumq(0, 0, 0) 0 >>> marcumq(0, a, 0) 1 - exp(-a**2/2) >>> marcumq(1, a, a) 1/2 + exp(-a**2)*besseli(0, a**2)/2 >>> marcumq(2, a, a) 1/2 + exp(-a**2)*besseli(0, a**2)/2 + exp(-a**2)*besseli(1, a**2) Differentiation with respect to $a$ and $b$ is supported: >>> from sympy import diff >>> diff(marcumq(m, a, b), a) a*(-marcumq(m, a, b) + marcumq(m + 1, a, b)) >>> diff(marcumq(m, a, b), b) -a**(1 - m)*b**m*exp(-a**2/2 - b**2/2)*besseli(m - 1, a*b) References ========== .. [1] https://en.wikipedia.org/wiki/Marcum_Q-function .. 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