MZdDddlmZmZmZddlmZddlmZmZddl m Z m Z m Z ddl mZmZmZddlmZddlmZmZmZddlmZmZmZdd lmZmZmZdd lm Z m!Z!m"Z"dd l#m$Z$dd l%m&Z&dd l'm(Z(m)Z)m*Z*m+Z+m,Z,m-Z-m.Z.m/Z/m0Z0m1Z1m2Z2ddl3m4Z4dZ5edZ6edZ7edZ8GddeZ9dZ:Gdde9Z;Gdde9Z<Gdde9Z=Gdde9Z>Gdde9Z?Gd d!e?Z@Gd"d#e?ZAGd$d%eZBGd&d'eBZCGd(d)eBZDGd*d+eBZEGd,d-eBZFGd.d/eBZGGd0d1eBZHy2)3)Ssympifycacheit)Add)FunctionArgumentIndexError)fuzzy_or fuzzy_and FuzzyBool)IpiRational)Dummy)binomial factorialRisingFactorial) bernoullieulernC)Absimre)explogmatch_real_imag)floor)sqrt) acosacotasinatancoscotcscsecsintan_imaginary_unit_as_coefficient)symmetric_polyc |j|jtDcic]}||jtc}Scc}wN)xreplaceatomsHyperbolicFunctionrewriter)exprhs G/usr/lib/python3/dist-packages/sympy/functions/elementary/hyperbolic.py_rewrite_hyperbolics_as_expr3sE ==./1 QYYs^+1 221sAc nitttdtdzzt tt dtdzztjt dz t ddt t ddztddz t dz td dz t t ddzdtdz t dz dtdz t t ddztddz t dz td dz t t ddztddz tdz t t dd ztddz tdz t t d d ztdtdzdz t dz tdtdz dz t t d dztdtdz dz t t ddztdtdz  dz t t ddzdtdzdtdzz t d z dtdz dtdzz t t d d ztddzdz t dz tddz dz t t ddziS) N  )r rrrHalfr rr2 _acosh_tablerCs  3q!d1g+   CAQK !  1  QHQN*   Q 2a4   a Bx1~%   $q' 2a4  47 Bx1~%  Q 2a4  a Bx1~%  a1d4j "Xa_"4  q'A+tDz!2hq"o#5  Qa[!RT  a$q'k 1b!Q/  Qa[!RA.  a$q'k 1b!Q/! " T!Wqay!2b5# $ d1g+$q' "BxB'7$7 a1aA q'A+q"Xa^+) rBc ltt dz ttdtdzzt dz tdtdzzt dz tdztdtdz z t dz tdzt dz ttddtdz zzt dz ttdzt dz ttddz zd tzdz tdztd z t d z tdztdtdzz d tzdz ttddtdz z zd tzdz ttdtdz zd tzdz tdt t dtdzdz zi S) Nr6r:r=r5r; r<r9r7)r r rrrrArBr2 _acsch_tablerI3sg sQw tAwa !B38 q47{ObS2X aC$q47{# #bS1W aC"q d1qay=! !B37 d1gIsQw tAwqyM2b52: aC$q'MB37 aC$q47{# #RUQY d1qay=! !2b519 tAwa !2b52: aD1"S!DG)Q''  rBchitttzdz tdtdzzt ttzdz tdtdzztdtdz tdz tdtdz dtzdz tddtdz z tdz tddtdz z  dtzdz dtdtdzz td z d tdtdzz d tzd z dtd z tdz d td z dtzdz tddz tdz dtdz d tzdz tdtd z td d tzd z tddtdz zd tzdz tddtdz z d tzdz t dtd z t d dtzd z tddtdzzd tzd z tddtdzz dtzd z dtdzdtzdz dtdz d tzdz tdtdzdtzdz td tdz d tzdz ttj zt tzdz ttj zttzdz i S)Nr6r5r:r=r?r;rE r<rGr>r7r9r8)r r rrrInfinityNegativeInfinityrArBr2 _asech_tablerNFs "Q$(|c!d1g+.. BAST!W-- !WtAw b !WtAw B  QtAwY b  !aQi- !B$)   Qa[! !26  a$q'k" "AbD1H  QKa  aL!B$( !Wq[26 a[1R4!8  GR!V !WHadQh  QtAwY 2 !aQi- !B$)! " aD"q&# $qTE1R4!8 AQK !1R4!8 !Qa[/ " "AbD1H a[1R4!8 $q'\AbD1H !WtAw 21gXQ !B$) ajjL2#a%!) a  "Q$(5  rBceZdZdZdZy)r.ze Base class for hyperbolic functions. See Also ======== sinh, cosh, tanh, coth TN)__name__ __module__ __qualname____doc__ unbranchedrArBr2r.r.jsJrBr.cNttz}tj|D]M}||k(rtj }nH|j s'|j\}}||k(s@|jsMn|tjfS|tjz}||z }|||zz |fS)a Split ARG into two parts, a "rest" and a multiple of $I\pi$. This assumes ARG to be an ``Add``. The multiple of $I\pi$ returned in the second position is always a ``Rational``. Examples ======== >>> from sympy.functions.elementary.hyperbolic import _peeloff_ipi as peel >>> from sympy import pi, I >>> from sympy.abc import x, y >>> peel(x + I*pi/2) (x, 1/2) >>> peel(x + I*2*pi/3 + I*pi*y) (x + I*pi*y + I*pi/6, 1/2) ) r r r make_argsrOneis_Mul as_two_terms is_RationalZeror@)argipiaKpm1m2s r2 _peeloff_ipircws" Q$C ]]3   8A  XX>>#DAqCxAMM AFF{ aff*B RB C< rBceZdZdZddZddZedZee dZ dZ ddZ ddZ dd Zdd Zd Zd ZdZdZdZdZdZddZdZdZdZdZdZdZy )sinha ``sinh(x)`` is the hyperbolic sine of ``x``. The hyperbolic sine function is $\frac{e^x - e^{-x}}{2}$. Examples ======== >>> from sympy import sinh >>> from sympy.abc import x >>> sinh(x) sinh(x) See Also ======== cosh, tanh, asinh cT|dk(rt|jdSt||)z@ Returns the first derivative of this function. r5r)coshargsrselfargindexs r2fdiffz sinh.fdiffs+ q= ! % %$T84 4rBctSz7 Returns the inverse of this function. asinhris r2inversez sinh.inverse  rBc|jr|tjurtjS|tjurtjS|tjurtjS|j rtj S|jr ||  Sy|tjurtjSt|}|tt|zS|jr ||  S|jrOt|\}}|r?|tztz}t!|t#|zt#|t!|zzS|j rtj S|j$t&k(r|j(dS|j$t*k(r,|j(d}t-|dz t-|dzzS|j$t.k(r#|j(d}|t-d|dzz z S|j$t0k(r/|j(d}dt-|dz t-|dzzz SyNrr5r6) is_NumberrNaNrLrMis_zeror[ is_negativeComplexInfinityr(r r&could_extract_minus_signis_Addrcr rergfuncrprhacoshratanhacoth)clsr\i_coeffxms r2evalz sinh.evals ==aee|uu  "zz!***)))vv SD z!!a'''uu 4S9G"3w<''//1I:%zz#C(1"QA747?T!WT!W_<<{{vv xx5 xx{"xx5 HHQKAE{T!a%[00xx5 HHQKa!Q$h''xx5 HHQK$q1u+QU 344!rBc|dks|dzdk(rtjSt|}t|dkDr|d}||dzz||dz zz S||zt |z S)zG Returns the next term in the Taylor series expansion. rr6rGr5rr[rlenrnrprevious_termsr`s r2 taylor_termzsinh.taylor_termsl q5AEQJ66M A>"Q&"2&1a4x1a!e9--1v ! ,,rBcZ|j|jdjSNrr|rh conjugaterjs r2_eval_conjugatezsinh._eval_conjugate"yy1//122rBc |jdjr<|r(d|d<|j|fi|tjfS|tjfS|r2|jdj|fi|j \}}n |jdj \}}t |t|zt|t|zfS)z@ Returns this function as a complex coordinate. rFcomplex rhis_extended_realexpandrr[ as_real_imagrer"rgr&rjdeephintsrrs r2rzsinh.as_real_imags 99Q< ( (#(i # D2E2AFF;;aff~% (TYYq\((77DDFFBYYq\..0FBRR $r(3r7"233rBc H|jdd|i|\}}||tzzSNrrArr rjrrre_partim_parts r2_eval_expand_complexzsinh._eval_expand_complex0,4,,@$@%@""rBc |r!|jdj|fi|}n|jd}d}|jr|j\}}nO|j d\}}|t j ur(|jr|t j ur |}|dz |z}|?t|tzt|t|zzjdSt|SNrTrationalr5)trig) rhrr{rY as_coeff_MulrrW is_Integerrergrjrrr\rycoefftermss r2_eval_expand_trigzsinh._eval_expand_trig %$))A,%%d4e4C))A,C  ::##%DAq++T+:LE5AEE!e&6&65;MQYM =GDGOd1gd1go5==4=H HCyrBNc 8t|t| z dz SNr6rrjr\limitvarkwargss r2_eval_rewrite_as_tractablezsinh._eval_rewrite_as_tractable&C3t9$))rBc 8t|t| z dz Srrrjr\rs r2_eval_rewrite_as_expzsinh._eval_rewrite_as_exp)rrBc 6t tt|zzSr+r r&rs r2_eval_rewrite_as_sinzsinh._eval_rewrite_as_sin,rCCL  rBc 6t tt|zz Sr+r r$rs r2_eval_rewrite_as_csczsinh._eval_rewrite_as_csc/rrBc Jt t|ttzdz zzSrr rgr rs r2_eval_rewrite_as_coshzsinh._eval_rewrite_as_cosh2 r$sRT!V|$$$rBc Vttj|z}d|zd|dzz z SNr6r5tanhrr@rjr\r tanh_halfs r2_eval_rewrite_as_tanhzsinh._eval_rewrite_as_tanh5s,$ {A 1 ,--rBc Vttj|z}d|z|dzdz z Srcothrr@rjr\r coth_halfs r2_eval_rewrite_as_cothzsinh._eval_rewrite_as_coth9s,$ {IqL1,--rBc dt|z SNr5cschrs r2_eval_rewrite_as_cschzsinh._eval_rewrite_as_csch=49}rBc*|jdj|||}|j|d}|tjur"|j |d|j rdnd}|jr|S|jr|j|S|SNrlogxcdir-+)dir) rhas_leading_termsubsrrvlimitrxrw is_finiter|rjrrrr\arg0s r2_eval_as_leading_termzsinh._eval_as_leading_term@siil**14d*Cxx1~ 155=99Qd.>.>sC9HD <<J ^^99T? "KrBc|jd}|jry|j\}}|tzjSNrTrhis_realrr rwrjr\rrs r2 _eval_is_realzsinh._eval_is_realMs;iil ;;!!#B2rBc8|jdjryyrrhrrs r2_eval_is_extended_realzsinh._eval_is_extended_realW 99Q< ( ( )rBch|jdjr|jdjSyrrhr is_positivers r2_eval_is_positivezsinh._eval_is_positive[, 99Q< ( (99Q<++ + )rBch|jdjr|jdjSyrrhrrxrs r2_eval_is_negativezsinh._eval_is_negative_rrBc8|jd}|jSrrhrrjr\s r2_eval_is_finitezsinh._eval_is_finiteciil}}rBcjt|jd\}}|jr |jSyr)rcrhrw is_integerrjrestipi_mults r2 _eval_is_zerozsinh._eval_is_zerogs0%diil3h <<&& & rBr5Tr+r)rPrQrRrSrlrq classmethodr staticmethodrrrrrrrrrrrrrrrrrrrrrrArBr2reres&5 .5.5`  -  -34 #"**!!%.. ,,'rBreceZdZdZddZedZeedZ dZ ddZ ddZ ddZ dd Zd Zd Zd ZdZdZdZdZddZdZdZdZdZdZy )rga" ``cosh(x)`` is the hyperbolic cosine of ``x``. The hyperbolic cosine function is $\frac{e^x + e^{-x}}{2}$. Examples ======== >>> from sympy import cosh >>> from sympy.abc import x >>> cosh(x) cosh(x) See Also ======== sinh, tanh, acosh cT|dk(rt|jdSt||Nr5rrerhrris r2rlz cosh.fdiffs) q= ! % %$T84 4rBcxddlm}|jr|tjurtjS|tj urtj S|tj urtj S|jrtjS|jr || Sy|tjurtjSt|}|||S|jr || S|jrOt|\}}|r?|tzt z}t#|t#|zt%|t%|zzS|jrtjS|j&t(k(rt+d|j,ddzzS|j&t.k(r|j,dS|j&t0k(r!dt+d|j,ddzz z S|j&t2k(r/|j,d}|t+|dz t+|dzzz Sy)Nr)r"r5r6)(sympy.functions.elementary.trigonometricr"rurrvrLrMrwrWrxryr(rzr{rcr r rgrer|rprrhr}r~r)rr\r"rrrs r2rz cosh.evals@ ==aee|uu  "zz!***zz!uu C4y !a'''uu 4S9G"7|#//1t9$zz#C(1"QA747?T!WT!W_<<{{uu xx5 A Q.//xx5 xx{"xx5 a#((1+q.0111xx5 HHQK$q1u+QU 344!rBc|dks|dzdk(rtjSt|}t|dkDr|d}||dzz||dz zz S||zt |z S)Nrr6r5rGrrs r2rzcosh.taylor_termsl q5AEQJ66M A>"Q&"2&1a4x1a!e9--1vil**rBcZ|j|jdjSrrrs r2rzcosh._eval_conjugaterrBc |jdjr<|r(d|d<|j|fi|tjfS|tjfS|r2|jdj|fi|j \}}n |jdj \}}t |t|zt|t|zfS)NrFr) rhrrrr[rrgr"rer&rs r2rzcosh.as_real_imags 99Q< ( (#(i # D2E2AFF;;aff~% (TYYq\((77DDFFBYYq\..0FBRR $r(3r7"233rBc H|jdd|i|\}}||tzzSrrrs r2rzcosh._eval_expand_complexrrBc |r!|jdj|fi|}n|jd}d}|jr|j\}}nO|j d\}}|t j ur(|jr|t j ur |}|dz |z}|?t|tzt|t|zzjdSt|Sr) rhrr{rYrrrWrrgrers r2rzcosh._eval_expand_trigrrBNc 8t|t| zdz Srrrs r2rzcosh._eval_rewrite_as_tractablerrBc 8t|t| zdz Srrrs r2rzcosh._eval_rewrite_as_exprrBc &tt|zSr+r"r rs r2_eval_rewrite_as_coszcosh._eval_rewrite_as_cos1s7|rBc ,dtt|zz Srr%r rs r2_eval_rewrite_as_seczcosh._eval_rewrite_as_sec3q3w<rBc Jt t|ttzdz zzSrr rer rs r2_eval_rewrite_as_sinhzcosh._eval_rewrite_as_sinhrrBc Vttj|zdz}d|zd|z z Srrrs r2rzcosh._eval_rewrite_as_tanhs,$a' I I ..rBc Vttj|zdz}|dz|dz z Srrrs r2rzcosh._eval_rewrite_as_coths,$a' A A ..rBc dt|z Srsechrs r2_eval_rewrite_as_sechzcosh._eval_rewrite_as_sechrrBcF|jdj|||}|j|d}|tjur"|j |d|j rdnd}|jrtjS|jr|j|S|Sr) rhrrrrvrrxrwrWrr|rs r2rzcosh._eval_as_leading_termsiil**14d*Cxx1~ 155=99Qd.>.>sC9HD <<55L ^^99T? "KrBc|jd}|js |jry|j\}}|tzj Sr)rhr is_imaginaryrr rwrs r2rzcosh._eval_is_realsEiil ;;#** !!#B2rBc |jd}|j\}}|dtzz}|j}|ry|j}|dur|St |t |t |tdz k|dtzdz kDgggSNrr6TFr7rhrr rwr r rjzrrymodyzeroxzeros r2rzcosh._eval_is_positives IIaL~~1AbDz    E>LdRTk4!B$q&=9:  rBc |jd}|j\}}|dtzz}|j}|ry|j}|dur|St |t |t |tdz k|dtzdz k\gggSr,r-r.s r2_eval_is_nonnegativezcosh._eval_is_nonnegative?s IIaL~~1AbDz    E>LdbdlDAbDFN;<  rBc8|jd}|jSrrrs r2rzcosh._eval_is_finiteYrrBct|jd\}}|r*|jr|tjz j Syyr)rcrhrwrr@rrs r2rzcosh._eval_is_zero]s=%diil3h  qvv%11 1%8rBrrr+r)rPrQrRrSrlrrr rrrrrrrrrrr!rrr'rrrr4rrrArBr2rgrgms&5 -5-5^  +  +3 4#"** %//  @42rBrgceZdZdZddZddZedZee dZ dZ ddZ dZ dd Zd Zd Zd ZdZdZdZddZdZdZdZdZdZdZy )ra' ``tanh(x)`` is the hyperbolic tangent of ``x``. The hyperbolic tangent function is $\frac{\sinh(x)}{\cosh(x)}$. Examples ======== >>> from sympy import tanh >>> from sympy.abc import x >>> tanh(x) tanh(x) See Also ======== sinh, cosh, atanh c||dk(r,tjt|jddzz St ||Nr5rr6)rrWrrhrris r2rlz tanh.fdiffws7 q=554 ! -q00 0$T84 4rBctSrnr~ris r2rqz tanh.inverse}rrrBc|jr|tjurtjS|tjurtjS|tj urtj S|jrtjS|jr ||  Sy|tjurtjSt|}|6|jrt t| zStt|zS|jr ||  S|jrQt!|\}}|rAt#|t$ztz}|tjur t'|St#|S|jrtjS|j(t*k(r#|j,d}|t/d|dzzz S|j(t0k(r/|j,d}t/|dz t/|dzz|z S|j(t2k(r|j,dS|j(t4k(rd|j,dz Syrt)rurrvrLrWrM NegativeOnerwr[rxryr(rzr r'r{rcrr rr|rprhrr}r~r)rr\rrrtanhms r2rz tanh.evals ==aee|uu  "uu ***}}$vv SD z!!a'''uu 4S9G"3352WH --3w<''//1I:%zz#C(1 2aLE 1 11#Aw#Aw{{vv xx5 HHQKa!Q$h''xx5 HHQKAE{T!a%[0144xx5 xx{"xx5 !}$!rBc|dks|dzdk(rtjSt|}d|dzz}t|dz}t |dz}||dz z|z|z ||zzSNrr6r5)rr[rrr)rrrr^BFs r2rztanh.taylor_termso q5AEQJ66M AAE A!a% A!a% Aa!e9q=?QT) )rBcZ|j|jdjSrrrs r2rztanh._eval_conjugaterrBc |jdjr<|r(d|d<|j|fi|tjfS|tjfS|r2|jdj|fi|j \}}n |jdj \}}t |dzt|dzz}t |t|z|z t|t|z|z fS)NrFrr6r)rjrrrrdenoms r2rztanh.as_real_imags 99Q< ( (#(i # D2E2AFF;;aff~% (TYYq\((77DDFFBYYq\..0FBR! c"gqj(Rb!%'RR)>??rBc |jd}|jrt|j}|jDcgc]}t|dj }}ddg}t |dzD]}||dzxxt ||z cc<|d|dz S|jr|j\}} |jr|dkDrt| } t d|dzdD cgc]} tt || | | zz}} t d|dzdD cgc]} tt || | | zz} } t|t| z St|Scc}wcc} wcc} w)NrFevaluater5r6) rhr{rrrranger)rXrrrr) rjrr\rrTXr`irrTkds r2rztanh._eval_expand_trigsbiil ::CHH A#q5);;=#B#AA1q5\ 2!a%N1b11 2Q4!9  ZZ++-LE5EAIK7J99S> !rBc|jd}|jry|j\}}|dk(r|tztdz k(ry|tdz zjS)NrTr6rrs r2rztanh._eval_is_real s[iil ;;!!#B 7rBw"Q$bd $$$rBc8|jdjryyrrrs r2rztanh._eval_is_extended_realrrBch|jdjr|jdjSyrrrs r2rztanh._eval_is_positiverrBch|jdjr|jdjSyrrrs r2rztanh._eval_is_negative"rrBc|jd}|j\}}t|dzt|dzz}|dk(ry|jry|j ryy)Nrr6FT)rhrr"re is_numberr)rjr\rrrEs r2rztanh._eval_is_finite&s`iil!!#BB T"Xq[( A: __    rBc<|jd}|jryyrrhrwrs r2rztanh._eval_is_zero2siil ;; rBrrr+r)rPrQrRrSrlrqrrr rrrrrrrrWrYr!rrrrrrrrrrArBr2rrcs&5  2%2%h  *  *3 @&77!!.." %,, rBrceZdZdZddZddZedZee dZ dZ ddZ dd Z d Zd Zd Zd ZdZdZddZdZy)ra+ ``coth(x)`` is the hyperbolic cotangent of ``x``. The hyperbolic cotangent function is $\frac{\cosh(x)}{\sinh(x)}$. Examples ======== >>> from sympy import coth >>> from sympy.abc import x >>> coth(x) coth(x) See Also ======== sinh, cosh, acoth c`|dk(rdt|jddzz St||)Nr5r8rr6r ris r2rlz coth.fdiffLs3 q=d499Q<(!++ +$T84 4rBctSrn)rris r2rqz coth.inverseRrrrBc|jr|tjurtjS|tjurtjS|tj urtj S|jrtjS|jr ||  Sy|tjurtjSt|}|6|jrtt| zSt t|zS|jr ||  S|jrQt|\}}|rAt!|t"ztz}|tjur t!|St%|S|jrtjS|j&t(k(r#|j*d}t-d|dzz|z S|j&t.k(r/|j*d}|t-|dz t-|dzzz S|j&t0k(rd|j*dz S|j&t2k(r|j*dSyrt)rurrvrLrWrMr=rwryrxr(rzr r#r{rcrr rr|rprhrr}r~r)rr\rrrcothms r2rz coth.evalXs ==aee|uu  "uu ***}}$(((SD z!!a'''uu 4S9G"335sG8},,rCL((//1I:%zz#C(1 2aLE 1 11#Aw#Aw{{(((xx5 HHQKA1H~a''xx5 HHQK$q1u+QU 344xx5 !}$xx5 xx{"!rBc|dk(rdt|z S|dks|dzdk(rtjSt|}t|dz}t |dz}d|dzz|z|z ||zzSrtrrr[rrrrrrArBs r2rzcoth.taylor_termsx 6wqz> ! 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N_is_even_is_oddc|jr+|jr || S|jr ||  S|jj |}t |dr"|j |k(r|jdS|d|z S|S)Nrqrr5)rzrr_reciprocal_ofrhasattrrqrh)rr\ts r2rz!ReciprocalHyperbolicFunction.evals  ' ' )||C4y {{SD z!    # #C ( 3 "s{{}';88A; mqs**rBcb|j|jd}t|||i|Sr)rrhgetattr)rj method_namerhros r2_call_reciprocalz-ReciprocalHyperbolicFunction._call_reciprocals3    ! -&wq+&777rBc@|j|g|i|}|d|z S|Sr)r)rjrrhrrs r2_calculate_reciprocalz2ReciprocalHyperbolicFunction._calculate_reciprocals3 "D ! !+ ? ? ?mqs**rBc`|j||}|||j|k7rd|z Syyr)rr)rjrr\rs r2_rewrite_reciprocalz0ReciprocalHyperbolicFunction._rewrite_reciprocals=  ! !+s 3 =Q$"5"5c"::Q3J;=rBc &|jd|S)Nrrrs r2rz1ReciprocalHyperbolicFunction._eval_rewrite_as_exp s''(>DDrBc &|jd|S)Nrrrs r2rz7ReciprocalHyperbolicFunction._eval_rewrite_as_tractables''(DcJJrBc &|jd|S)Nrrrs r2rz2ReciprocalHyperbolicFunction._eval_rewrite_as_tanh''(?EErBc &|jd|S)Nrrrs r2rz2ReciprocalHyperbolicFunction._eval_rewrite_as_cothrrBc fd|j|jdz j|fi|Sr )rrhr)rjrrs r2rz)ReciprocalHyperbolicFunction.as_real_imags2CD'' ! 55CCDRERRrBcZ|j|jdjSrrrs r2rz,ReciprocalHyperbolicFunction._eval_conjugaterrBc H|jdddi|\}}|t|zzS)NrTrArrs r2rz1ReciprocalHyperbolicFunction._eval_expand_complexs0,4,,@$@%@7""rBc &|jdi|S)N)r)r)rjrs r2rz.ReciprocalHyperbolicFunction._eval_expand_trig!s)t))GGGrBcbd|j|jdz j|Sr )rrhr)rjrrrs r2rz2ReciprocalHyperbolicFunction._eval_as_leading_term$s+$%%diil33JJ1MMrBcR|j|jdjSr)rrhrrs r2rz3ReciprocalHyperbolicFunction._eval_is_extended_real's!""499Q<0AAArBcXd|j|jdz jSr )rrhrrs r2rz,ReciprocalHyperbolicFunction._eval_is_finite*s&$%%diil33>>>rBr+rr)rPrQrRrSrrr __annotations__rrrrrrrrrrrrrrrrrrArBr2rrsGNHiGY + +8 + EKFFS3#HNB?rBrc^eZdZdZeZdZd dZee dZ dZ dZ dZ dZd Zd Zy ) ra8 ``csch(x)`` is the hyperbolic cosecant of ``x``. The hyperbolic cosecant function is $\frac{2}{e^x - e^{-x}}$ Examples ======== >>> from sympy import csch >>> from sympy.abc import x >>> csch(x) csch(x) See Also ======== sinh, cosh, tanh, sech, asinh, acosh Tc|dk(r2t|jd t|jdzSt||)z? Returns the first derivative of this function r5r)rrhrrris r2rlz csch.fdiffEs@ q=1&&diil);; ;$T84 4rBc|dk(rdt|z S|dks|dzdk(rtjSt|}t|dz}t |dz}ddd|zz z|z|z ||zzS)zF Returns the next term in the Taylor series expansion rr5r6rurvs r2rzcsch.taylor_termNs} 6WQZ<  Ua!eqj66M A!a% A!a% AAqD>A%a'!Q$. .rBc 4ttt|zz Sr+rrs r2rzcsch._eval_rewrite_as_sin`3q3w<rBc 4ttt|zzSr+rrs r2rzcsch._eval_rewrite_as_csccrrBc Htt|ttzdz zz Srrrs r2rzcsch._eval_rewrite_as_coshfs4a"fqj()))rBc dt|z Srrers r2r!zcsch._eval_rewrite_as_sinhirrBch|jdjr|jdjSyrrrs r2rzcsch._eval_is_positivelrrBch|jdjr|jdjSyrrrs r2rzcsch._eval_is_negativeprrBNr)rPrQrRrSrerrrlr rrrrrr!rrrArBr2rr.sR&NG5 / /   *,,rBrcXeZdZdZeZdZd dZee dZ dZ dZ dZ dZd Zy ) r&a: ``sech(x)`` is the hyperbolic secant of ``x``. The hyperbolic secant function is $\frac{2}{e^x + e^{-x}}$ Examples ======== >>> from sympy import sech >>> from sympy.abc import x >>> sech(x) sech(x) See Also ======== sinh, cosh, tanh, coth, csch, asinh, acosh Tc|dk(r2t|jd t|jdzSt||r )rrhr&rris r2rlz sech.fdiffs> q=$))A,''TYYq\(:: :$T84 4rBc|dks|dzdk(rtjSt|}t|t |z ||zzSr@)rr[rrrrrrs r2rzsech.taylor_termsC q5AEQJ66M A8il*QV3 3rBc ,dtt|zz Srrrs r2rzsech._eval_rewrite_as_cosrrBc &tt|zSr+rrs r2rzsech._eval_rewrite_as_secrrBc Htt|ttzdz zz Srr rs r2r!zsech._eval_rewrite_as_sinhs4a"fai(((rBc dt|z Srrgrs r2rzsech._eval_rewrite_as_coshrrBc8|jdjryyrrrs r2rzsech._eval_is_positiverrBNr)rPrQrRrSrgrrrlr rrrrr!rrrArBr2r&r&usM&NH5  4 4 )rBr&ceZdZdZy)InverseHyperbolicFunctionz,Base class for inverse hyperbolic functions.N)rPrQrRrSrArBr2rrs6rBrceZdZdZddZedZeedZ ddZ ddZ dZ e Z d Zd Zd Zd Zdd ZdZy)rpaM ``asinh(x)`` is the inverse hyperbolic sine of ``x``. The inverse hyperbolic sine function. Examples ======== >>> from sympy import asinh >>> from sympy.abc import x >>> asinh(x).diff(x) 1/sqrt(x**2 + 1) >>> asinh(1) log(1 + sqrt(2)) See Also ======== acosh, atanh, sinh cf|dk(r!dt|jddzdzz St||r9)rrhrris r2rlz asinh.fdiffs7 q=T$))A,/A-.. .$T84 4rBc&|jr|tjurtjS|tjurtjS|tjurtjS|j rtj S|tjurttddzS|tjurttddz S|jr||  S|tjurtjS|j rtj St|}|tt|zS|j!r ||  St#|t$r|j&dj(rz|j&d}|j*r|St-|\}}|L|It/|t0dz zt0z }|tt0z|zz }|j2}|dur|S|dur| Syyyyy)Nr6r5rTF)rurrvrLrMrwr[rWrrr=rxryr(r r rz isinstancererhrlrrrr is_even) rr\rr/rrKfrevens r2rz asinh.evals ==aee|uu  "zz!***)))vv 47Q;'' %47Q;''SD z!a'''((({{vv 4S9G"4=((//1I:% c4 SXXa[%:%: Ayy"1%DAq}1r!t8R-("QJyy4<HU]2I# "/} &; rBcV|dks|dzdk(rtjSt|}t|dk\r%|dkDr |d}| |dz dzz||dz zz |dzzS|dz dz}t tj |}t |}tj|z|z|z ||zz|z SNrr6rGr5)rr[rrrr@rr=rrrr`rMRrBs r2rzasinh.taylor_terms q5AEQJ66M A>"a'AE"2&rQUQJ1q5 2QT99UqL#AFFA.aL}}a'!+a/!Q$6::rBNc |jd}|j|dj}|jr|j |S|t t t jfvr'|jtj|||Sd|dzzjr|j||r|nd}t|jr5t|jr|j!| t t"zz St|jr5t|jrG|j!| t t"zzS|jtj|||S|j!|SNrrr5r6)rhrcancelrwrr rryr/rrrxrrrrr|r rjrrrr\x0ndirs r2rzasinh._eval_as_leading_terms2iil XXa^ " " $ ::&&q) ) 1"a**+ +<<$::14d:S S AI " "771dd2D$x##b6%% IIbM>AbD00D%%b6%% IIbM>AbD00||C(>>qtRV>WWyy}rBc|jd}|j|d}|tt fvr(|jtj ||||St j ||||}|tjur|Sd|dzzjr|j||r|nd}t|jr(t|jr| ttzz S|St|jr(t|jr| ttzzS|S|jtj ||||S|SNrrrrr5r6)rhrr r/r _eval_nseriesrrryrxrrrrr rjrrrrr\rresrs r2rzasinh._eval_nseries%s.iilxx1~ Ar7?<<$221ad2N N$$T1= 1$$ $J aK $ $771dd2D$x##d8''4!B$;&  D%%d8''4!B$;& ||C(66q!$T6RR rBc <t|t|dzdzzSrrrrjrrs r2_eval_rewrite_as_logzasinh._eval_rewrite_as_log>s1tAqD1H~%&&rBc <t|td|dzzz SNr5r6)r~rrs r2_eval_rewrite_as_atanhzasinh._eval_rewrite_as_atanhCsQtA1H~%&&rBc t|z}ttd|z t|dz z t|ztdz z zSr)r rr}r )rjrrixs r2_eval_rewrite_as_acoshzasinh._eval_rewrite_as_acoshFs= qS$q2v,tBF|+eBi7"Q$>??rBc 6t tt|zzSr+)r r rs r2_eval_rewrite_as_asinzasinh._eval_rewrite_as_asinJsrDQKrBc Vttt|zzttzdz z Sr)r rr rs r2_eval_rewrite_as_acoszasinh._eval_rewrite_as_acosMs!4A;2a''rBctSrnrris r2rqz asinh.inverseP  rBc4|jdjSrrnrs r2rzasinh._eval_is_zeroVsyy|###rBrrr)rPrQrRrSrlrrr rrrrrrrrrrrqrrArBr2rprpss*5 ++Z  ;  ;*2'"6'@ ( $rBrpceZdZdZddZedZeedZ ddZ ddZ dZ e Z d Zd Zd Zd Zdd ZdZy)r}aM ``acosh(x)`` is the inverse hyperbolic cosine of ``x``. The inverse hyperbolic cosine function. Examples ======== >>> from sympy import acosh >>> from sympy.abc import x >>> acosh(x).diff(x) 1/(sqrt(x - 1)*sqrt(x + 1)) >>> acosh(1) 0 See Also ======== asinh, atanh, cosh c|dk(r/|jd}dt|dz t|dzzz St||r rhrr)rjrkr\s r2rlz acosh.fdiffpsC q=))A,Cd37mDqM12 2$T84 4rBcj|jr|tjurtjS|tjurtjS|tjurtjS|j rt tzdz S|tjurtjS|tjur t tzS|jr+t}||vr|jr ||tzS||S|tjurtjS|ttjzk(r!tjtt zdz zS|t tjzk(r!tjtt zdz z S|j rt tztjzSt!|t"r|j$djr|j$d}|j&r t)|St+|\}}||t-|t z }|tt z|zz }|j.}|dur|j0r|S|j2r| Sy|dur.|tt zz}|j4r| S|j6r|Syyyyyy)Nr6rTF)rurrvrLrMrwr r rWr[r=rlrCrryr@rrgrhrrrrris_nonnegativerxis_nonpositiver) rr\ cst_tabler/rrKrrrs r2rz acosh.evalws) ==aee|uu  "zz!***zz!!taxvv  %!t ==$Ii''$S>!++ ~% !## #$$ $ !AJJ, ::"Q& & 1"QZZ- ::"Q& & ;;a4;  c4 SXXa[%:%: Ayy1v "1%DAq}!B$K"QJyy4<''  !r 'U]2IA'' !r  ' #"/} &; rBcf|dk(rttzdz S|dks|dzdk(rtjSt |}t |dk\r$|dkDr|d}||dz dzz||dz zz |dzzS|dz dz}t tj|}t|}| |z tz||zz|z Sr) r r rr[rrrr@rrs r2rzacosh.taylor_terms 6R46M Ua!eqj66M A>"a'AE"2&AEA:~q!a%y1AqD88UqL#AFFA.aLrAvzAqD(1,,rBNc|jd}|j|dj}|tj tj tjtj fvr'|jtj|||S|dz jr|j||r|nd}t|jrC|dzjr"|j|dtztzz S|j| St|j s'|jtj|||S|j|Sr)rhrrrrWr[ryr/rrrxrrr|r r rrs r2rzacosh._eval_as_leading_termsiil XXa^ " " $ 155&!&&!%%):):; ;<<$::14d:S S F  771dd2D$x##F''99R=1Q3r611 " ~%X))||C(>>qtRV>WWyy}rBcn|jd}|j|d}|tjtjfvr(|j t j||||Stj||||}|tjur|S|dz jr|j||r|nd}t|jr%|dzjr|dtztzz S| St|js(|j t j||||S|SrrhrrrWr=r/rrrryrxrrr r rrs r2rzacosh._eval_nseriess iilxx1~ AEE1==) )<<$221ad2N N$$T1= 1$$ $J 1H ! !771dd2D$x##1H))1R<'t X))||C(66q!$T6RR rBc Tt|t|dzt|dz zzSrrrs r2rzacosh._eval_rewrite_as_logs'1tAE{T!a%[0011rBc Tt|dz td|z z t|zSr)rrrs r2rzacosh._eval_rewrite_as_acoss&AE{4A;&a00rBc ht|dz td|z z tdz t|z zSr)rr r rs r2rzacosh._eval_rewrite_as_asins.AE{4A;&"Q$a.99rBc t|dz td|z z tdz ttt|zzzzSr)rr r rprs r2_eval_rewrite_as_asinhzacosh._eval_rewrite_as_asinhs7AE{4A;&"Q$51:*=>>rBc t|dz }td|z }t|dzdz }tdz |z|z d|td|dzz zz z|t|dzz|z t||z zzSr)rr r~)rjrrsxm1s1mxsx2m1s r2rzacosh._eval_rewrite_as_atanhsAE{AE{QTAX1T $AQq!tV $4 45T!a%[ &uQw78 9rBctSrnrris r2rqz acosh.inverserrBc>|jddz jryy)Nrr5Trnrs r2rzacosh._eval_is_zeros IIaL1  % % &rBrrr)rPrQrRrSrlrrr rrrrrrrrrrrqrrArBr2r}r}Zsr*54!4!l - - ".2"61:?9 rBr}cveZdZdZd dZedZeedZ ddZ ddZ dZ e Z d Zd Zd Zd d Zy)r~a) ``atanh(x)`` is the inverse hyperbolic tangent of ``x``. The inverse hyperbolic tangent function. Examples ======== >>> from sympy import atanh >>> from sympy.abc import x >>> atanh(x).diff(x) 1/(1 - x**2) See Also ======== asinh, acosh, tanh cT|dk(rdd|jddzz z St||r9rhrris r2rlz atanh.fdiff2 q=a$))A,/)* *$T84 4rBcX|jr|tjurtjS|jrtjS|tj urtj S|tjurtjS|tj urt t|zS|tjurtt| zS|jrz||  S|tjur%ddl m}t|t dz tdz zSt!|}|tt|zS|j#r ||  S|jrtjSt%|t&r|j(dj*r|j(d}|j,r|St/|\}}|X|Ut1d|ztz }|j2}|t|ztzdz z } |dur| S|dur| ttzdz z Syyyyy)Nr AccumBoundsr6TF)rurrvrwr[rWrLr=rMr r!rxry!sympy.calculus.accumulationboundsrr r(rzrrrhrlrrrr) rr\rrr/rrKrrrs r2rz atanh.eval"s ==aee|uu vv zz! %))) "rDI~%***4:~%SD z!a'''IbSUBqD1114S9G"4=((//1I:% ;;66M c4 SXXa[%:%: Ayy"1%DAq}!A#b&Myy!BqL4<HU]qtAv:%# "/} &; rBcb|dks|dzdk(rtjSt|}||z|z SNrr6)rr[rrs r2rzatanh.taylor_termQs4 q5AEQJ66M Aa4!8OrBNc |jd}|j|dj}|jr|j |S|t j t j t jfvr'|jtj|||Sd|dzz jr|j||r|nd}t|jr+|jr|j|tt zz St|j"r+|j"rF|j|tt zzS|jtj|||S|j|Sr)rhrrrwrrrWryr/rrrxrrr|r r rrs r2rzatanh._eval_as_leading_termZs*iil XXa^ " " $ ::&&q) ) 155&!%%!2!23 3<<$::14d:S S AI " "771dd2D$x##>>99R=1R4//D%%>>99R=1R4//||C(>>qtRV>WWyy}rBc|jd}|j|d}|tjtjfvr(|j t j||||Stj||||}|tjur|Sd|dzz jr|j||r|nd}t|jr|jr|ttzz S|St|jr|jr|ttzzS|S|j t j||||S|Srrrs r2rzatanh._eval_nseriesos*iilxx1~ AEE1==) )<<$221ad2N N$$T1= 1$$ $J aK $ $771dd2D$x####2:%  D%%##2:% ||C(66q!$T6RR rBc Btd|ztd|z z dz Srrrs r2rzatanh._eval_rewrite_as_logs"AE SQZ'1,,rBc td|dzdz z }t|zdt|dz zz t| td|dzz zt|z |zt|zz Sr)rr rp)rjrrrs r2rzatanh._eval_rewrite_as_asinhsm AqD1H 1aadU m$aRa!Q$h'Q/1%(:; >> from sympy import acoth >>> from sympy.abc import x >>> acoth(x).diff(x) 1/(1 - x**2) See Also ======== asinh, acosh, coth cT|dk(rdd|jddzz z St||r9rris r2rlz acoth.fdiffrrBc|jr|tjurtjS|tjurtjS|tj urtjS|j rttzdz S|tjurtjS|tjurtj S|jrf||  S|tjurtjSt|}|t t|zS|jr ||  S|j rttztj zSyr)rurrvrLr[rMrwr r rWr=rxryr(rrzr@)rr\rs r2rz acoth.evals ==aee|uu  "vv ***vv !taxzz! %)))SD z!a'''vv 4S9G"rDM))//1I:% ;;a4;  rBc|dk(rt tzdz S|dks|dzdk(rtjSt |}||z|z Sr)r r rr[rrs r2rzacoth.taylor_termsJ 62b57N Ua!eqj66M Aa4!8OrBNc6|jd}|j|dj}|tjurd|z j |S|tj tj tjfvr'|jtj|||S|jrd|dzz jr|j||r|nd}t|jr+|jr|j!|t"t$zzSt|jr+|jrF|j!|t"t$zz S|jtj|||S|j!|S)Nrr5rr6)rhrrrryrrWr[r/rrrrrrrxr|r r rs r2rzacoth._eval_as_leading_terms9iil XXa^ " " $ "" "cE**1- - 155&!%%( (<<$::14d:S S ::1r1u911771dd2D$x##>>99R=1R4//D%%>>99R=1R4//||C(>>qtRV>WWyy}rBc|jd}|j|d}|tjtjfvr(|j t j||||Stj||||}|tjur|S|jrd|dzz jr|j||r|nd}t|jr|jr|tt zzS|St|jr|jr|tt zz S|S|j t j||||S|Sr)rhrrrWr=r/rrrryrrrrrxr r rs r2rzacoth._eval_nseriess0iilxx1~ AEE1==) )<<$221ad2N N$$T1= 1$$ $J <  *21"6:rBrc|eZdZdZddZedZeedZ ddZ ddZ ddZ d Z e Zd Zd Zd Zd Zy)asecha ``asech(x)`` is the inverse hyperbolic secant of ``x``. The inverse hyperbolic secant function. Examples ======== >>> from sympy import asech, sqrt, S >>> from sympy.abc import x >>> asech(x).diff(x) -1/(x*sqrt(1 - x**2)) >>> asech(1).diff(x) 0 >>> asech(1) 0 >>> asech(S(2)) I*pi/3 >>> asech(-sqrt(2)) 3*I*pi/4 >>> asech((sqrt(6) - sqrt(2))) I*pi/12 See Also ======== asinh, atanh, cosh, acoth References ========== .. [1] https://en.wikipedia.org/wiki/Hyperbolic_function .. [2] https://dlmf.nist.gov/4.37 .. [3] https://functions.wolfram.com/ElementaryFunctions/ArcSech/ cp|dk(r&|jd}d|td|dzz zz St||Nr5rr8r6rrjrkr/s r2rlz asech.fdiffLs? q= ! Aqa!Q$h'( ($T84 4rBc|jr|tjurtjS|tjurtt zdz S|tj urtt zdz S|jrtjS|tjurtjS|tjur tt zS|jr+t}||vr|jr ||t zS||S|tjur%ddlm}t |t dz tdz zS|jrtjSy)Nr6rr)rurrvrLr r rMrwrWr[r=rlrNrryrr)rr\rrs r2rz asech.evalSs ==aee|uu  "!tax***!taxzz!vv  %!t ==$Ii''$S>!++ ~% !## # E["Q1-- - ;;::  rBc||dk(rtd|z S|dks|dzdk(rtjSt|}t |dkDr*|dkDr%|d}||dz |dz zz|dzzd|dzdzzz S|dz}t tj ||z}t||zdz|zdz}d|z|z ||zzdz S)Nrr6r5rGr9r8)rrr[rrrr@rrs r2rzasech.taylor_termrs 6q1u:  Ua!eqj66M A>"Q&1q5"2&QUQqSM*QT111qy=AAF#AFFA.2aL1$)A-2AvzAqD(1,,rBNc|jd}|j|dj}|tj tj tjtj fvr'|jtj|||S|jsd|z jr|j||r|nd}t|jrO|js|dzjr|j| S|j|dtzt zz St|js'|jtj|||S|j|Sr)rhrrrrWr[ryr/rrrxrrrr|r r rs r2rzasech._eval_as_leading_termsiil XXa^ " " $ 155&!&&!%%):):; ;<<$::14d:S S >>a"f11771dd2D$x##>>b1f%9%9 IIbM>)yy}qs2v--X))||C(>>qtRV>WWyy}rBcddlm}|jd}|j|d}|tj ur]t dd}ttj |dzz jtj|dd|z} tj |jdz } | j|} | | z | z } | j|ds|dk(r|dS|t|Sttj | zj|||} | jt| zj!}| jj||j!j#|||z|zS|tj$urkt dd}ttj$|dzzjtj|dd|z} tj |jdz} | j|} | | z | z } | j|ds,|dk(r|dSt&t(z|t|zSttj | zj|||} | jt| zj!}| jj||j!j#|||z|zSt+j||||}|tj,ur|S|j.sd|z j.r|j1||r|nd}t3|j4r1|j4s|dzj.r| S|dt&zt(zz St3|j.s(|jtj|||| S|S Nr)OrT)positiver6r5rr)rcr#rhrrrWrrr/rnseriesris_meromorphicrrremoveOrpowsimpr=r r rryrxrrrrjrrrrr#r\rrserarg1rgres1rrs r2rzasech._eval_nseriess<(iilxx1~ 155=cD)A1 %--c2::1a1EC55499Q<'D$$Q'AA A##Aq) Avqt51T!W:5 ?00ad0CD<<>$q')113C;;=%%a-446>>@1QT1:M M 1== cD)A 1,-55c:BB1a1MC55499Q<'D$$Q'AA A##Aq) Avqt<1R4!DG*+<< ?00ad0CD<<>$q')113C;;=%%a-446>>@1QT1:M M$$T1= 1$$ $J   D55771dd2D$x####q'='=4KQqSV|#X))||C(66q!$T6RR rBctSrnr%ris r2rqz asech.inverserrBc ftd|z td|z dz td|z dzzzSrrrs r2rzasech._eval_rewrite_as_logs31S54# ?T!C%!)_<<==rBc td|z Sr)r}rs r2rzasech._eval_rewrite_as_acoshQsU|rBc td|z dz tdd|z z z ttt|z zttj zzzSr)rr rpr rr@rs r2rzasech._eval_rewrite_as_asinhsJAcEAItA#I.%#,24QVV)1<= =rBc httzdt|td|z zz tdz t| zt|z z tdz t|dzzt|dz z z ztd|dzz t|dzzttd|dzz zzSr)r r rr~rs r2rzasech._eval_rewrite_as_atanhs"a$q'$qs)++ac$r(l47.BBQqSaQRd^TXZ[]^Z^Y^T_E__`q!a%y/$q1u+-eDQTN.CCD ErBc td|z dz tdd|z z z tdz ttt|zzz zSr)rr r acschrs r2_eval_rewrite_as_acschzasech._eval_rewrite_as_acschs?AaC!G}T!ac']*BqD1U1Q3Z<,?@@rBrrr)rPrQrRrSrlrrr rrrrrqrrrrrr6rArBr2rr&sp#J5< - - "+Z >"6=EArBrc|eZdZdZddZedZeedZ ddZ ddZ ddZ d Z e Zd Zd Zd Zd Zy)r5a ``acsch(x)`` is the inverse hyperbolic cosecant of ``x``. The inverse hyperbolic cosecant function. Examples ======== >>> from sympy import acsch, sqrt, I >>> from sympy.abc import x >>> acsch(x).diff(x) -1/(x**2*sqrt(1 + x**(-2))) >>> acsch(1).diff(x) 0 >>> acsch(1) log(1 + sqrt(2)) >>> acsch(I) -I*pi/2 >>> acsch(-2*I) I*pi/6 >>> acsch(I*(sqrt(6) - sqrt(2))) -5*I*pi/12 See Also ======== asinh References ========== .. [1] https://en.wikipedia.org/wiki/Hyperbolic_function .. [2] https://dlmf.nist.gov/4.37 .. [3] https://functions.wolfram.com/ElementaryFunctions/ArcCsch/ c||dk(r,|jd}d|dztdd|dzz zzz St||rrrs r2rlz acsch.fdiffsH q= ! 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"cD)AQT "**3/771acBCtyy|#D$$Q'AA A##Aq) Avqt>1R46Ad1gJ+>> ?00ad0CD<<>$q')113C;;=%%a-446>>@1QT1:M M$$T1= 1$$ $J   !dAg+!:!:99Q<##Att;D$x##d8''4!B$;&  D%%d8''4!B$;& ||C(66q!$T6RR rBctSrnrris r2rqz acsch.inverserrBc Htd|z td|dzz dzzSrrrs r2rzacsch._eval_rewrite_as_logs'1S54#q&1 --..rBc td|z Srrors r2rzacsch._eval_rewrite_as_asinhr1rBc ttdt|z z tt|z dz z tt|z zttj zz zSr)r rr}r rr@rs r2rzacsch._eval_rewrite_as_acoshsN$q1S5y/$quqy/1 %ae -/1!&&y9: :rBc |dz}|dz}t| |z ttjzt|dz |z t t|zz zSr)rr rr@r~)rjr\rarg2arg2p1s r2rzacsch._eval_rewrite_as_atanhs^AvTE{3166 $faiZ 0 7d6l8K K!LM MrBc4|jdjSr)rhr:rs r2rzacsch._eval_is_zerosyy|'''rBrrr)rPrQrRrSrlrrr rrrrrqrrrrrrrArBr2r5r5sr#J5B @ @ *.` /"6:M (rBr5N)I sympy.corerrrsympy.core.addrsympy.core.functionrrsympy.core.logicr r r sympy.core.numbersr r rsympy.core.symbolr(sympy.functions.combinatorial.factorialsrrr%sympy.functions.combinatorial.numbersrrr$sympy.functions.elementary.complexesrrr&sympy.functions.elementary.exponentialrrr#sympy.functions.elementary.integersr(sympy.functions.elementary.miscellaneousrrrrr r!r"r#r$r%r&r'r(sympy.polys.specialpolysr)r3rCrIrNr.rcrergrrrrr&rrpr}r~rrr5rArBr2rSs**<;;..#GGFF<<LL59$$$$42   2    $    F  DQ' Q'hs2 s2lR Rji iXG?#5G?TD, 'D,N4 '4v  a$ %a$Hk %k\U %UpC %CLsA %sAl( %(rB