o GZh6@sdZddlmZddlmZddlmZddlmZm Z m Z ddl m Z ddl mZmZmZmZddlmZdd lmZdd lmZdd lmZmZdd lmZdd lmZmZm Z ddl!m"Z"m#Z#m$Z$ddl%m&Z&m'Z'ddl(m)Z)m*Z*ddl+m,Z,m-Z-m.Z.ddl/m0Z0m1Z1ddl2m3Z3m4Z4m5Z5ddl6m7Z7m8Z8dEddZ9GdddeZ:GdddeZ;GdddeZGd"d#d#eZ?Gd$d%d%eZ@Gd&d'd'eZAGd(d)d)eZBd*d+ZCGd,d-d-eZDGd.d/d/eZEGd0d1d1eZFGd2d3d3eFZGGd4d5d5eFZHGd6d7d7eFZIGd8d9d9eFZJGd:d;d;eZKGdd?d?eKZMGd@dAdAeZNGdBdCdCeZOdDS)Fz This module contains various functions that are special cases of incomplete gamma functions. It should probably be renamed. ) EulerGamma)Add)cacheit)DefinedFunctionArgumentIndexError expand_mul)fuzzy_or)IpiRationalInteger)is_eq)Pow)S)Dummyuniquely_named_symbol)sympify) factorial factorial2RisingFactorial) polar_liftre unpolarify)ceilingfloor)sqrtroot)explog exp_polar)coshsinh)cossinsinc)hypermeijergTcKs|jdjr|rd|d<|j|fi|tjfS|tjfS|r0|jdj|fi|\}}n |jd\}}||t|||t|d}||t|||t|dt}||fS)NrFcomplex)argsis_extended_realexpandrZero as_real_imagfuncr )selfdeephintsxyrZimr4V/var/www/auris/lib/python3.10/site-packages/sympy/functions/special/error_functions.pyreal_to_real_as_real_imags  "(,r6cseZdZdZdZd1ddZd1ddZedd Ze e d d Z d d Z ddZ ddZddZddZddZddZddZddZddZd d!Zd"d#Zd$d%Zd2d'd(Zd)d*Zd+d,Zd-d.Zfd/d0ZeZZ S)3erfa. The Gauss error function. Explanation =========== This function is defined as: .. math :: \mathrm{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} \mathrm{d}t. Examples ======== >>> from sympy import I, oo, erf >>> from sympy.abc import z Several special values are known: >>> erf(0) 0 >>> erf(oo) 1 >>> erf(-oo) -1 >>> erf(I*oo) oo*I >>> erf(-I*oo) -oo*I In general one can pull out factors of -1 and $I$ from the argument: >>> erf(-z) -erf(z) The error function obeys the mirror symmetry: >>> from sympy import conjugate >>> conjugate(erf(z)) erf(conjugate(z)) Differentiation with respect to $z$ is supported: >>> from sympy import diff >>> diff(erf(z), z) 2*exp(-z**2)/sqrt(pi) We can numerically evaluate the error function to arbitrary precision on the whole complex plane: >>> erf(4).evalf(30) 0.999999984582742099719981147840 >>> erf(-4*I).evalf(30) -1296959.73071763923152794095062*I See Also ======== erfc: Complementary error function. erfi: Imaginary error function. erf2: Two-argument error function. erfinv: Inverse error function. erfcinv: Inverse Complementary error function. erf2inv: Inverse two-argument error function. References ========== .. [1] https://en.wikipedia.org/wiki/Error_function .. [2] https://dlmf.nist.gov/7 .. [3] https://mathworld.wolfram.com/Erf.html .. [4] https://functions.wolfram.com/GammaBetaErf/Erf TcCs2|dkrdt|jdd ttSt||Nr8r(rrr)rr rr/argindexr4r4r5fdiff  z erf.fdiffcCtSz8 Returns the inverse of this function. erfinvr;r4r4r5inversez erf.inversecCs|jr!|tjur tjS|tjurtjS|tjurtjS|jr!tjSt |t r+|j dSt |t r8tj|j dS|jr>tjSt |t rN|j djrN|j dS|t}|tjtjfvr]|S|rg||  SdSNrr8) is_NumberrNaNInfinityOneNegativeInfinity NegativeOneis_zeror, isinstancerBr)erfcinverf2invextract_multiplicativelyr could_extract_minus_signclsargtr4r4r5evals.         zerf.evalcGs|dks |ddkr tjSt|}t|dtd}t|dkr2|d |d|d||Sdtj||||t|ttSNrr(r8) rr,rrlenrKrrr nr2previous_termskr4r4r5 taylor_terms "*zerf.taylor_termcC||jdSNrr.r) conjugater/r4r4r5_eval_conjugatezerf._eval_conjugatecC|jdjdur dSdSNrTr)r*rcr4r4r5 _eval_is_realzerf._eval_is_realcCrfrg)r) is_imaginaryrcr4r4r5_eval_is_imaginaryrjzerf._eval_is_imaginarycCs|jd}t|j|jgSr`)r)r is_finiter*r/zr4r4r5_eval_is_finites zerf._eval_is_finitecC |jdjdur|jdjSdSrg)r)r*rLrcr4r4r5 _eval_is_zero zerf._eval_is_zerocCrqrg)r)r*Zis_extended_positivercr4r4r5_eval_is_positiverszerf._eval_is_positivecCrqrg)r)r*is_extended_negativercr4r4r5_eval_is_negativerszerf._eval_is_negativecKs:ddlm}t|d|tj|tj|dttSNr uppergammar('sympy.functions.special.gamma_functionsryrrrIHalfr r/rokwargsryr4r4r5_eval_rewrite_as_uppergammas .zerf._eval_rewrite_as_uppergammacK4tjt|tt}tjtt|tt|SNrrIr rr fresnelcfresnelsr/ror~rTr4r4r5_eval_rewrite_as_fresnelszerf._eval_rewrite_as_fresnelscKrrrrr4r4r5_eval_rewrite_as_fresnelcrzerf._eval_rewrite_as_fresnelccKs.|ttttjggdgtddg|dSNrr(rr r&rr|r r/ror~r4r4r5_eval_rewrite_as_meijerg.zerf._eval_rewrite_as_meijergcKs.d|ttttjgdtjg|d SNr(rr r%rr|rr4r4r5_eval_rewrite_as_hyperrzerf._eval_rewrite_as_hypercKs,t|d||ttj|dttSNr(rexpintrr|r rr4r4r5_eval_rewrite_as_expint,zerf._eval_rewrite_as_expintNcKsbddlm}|r#|||tj}|tjur#tjt| t|d Stjt|t|d S)Nr)limitr() Zsympy.series.limitsrrrHrJrK_erfsrrI)r/rolimitvarr~rZlimr4r4r5_eval_rewrite_as_tractables  zerf._eval_rewrite_as_tractablecKtjt|Sr)rrIerfcrr4r4r5_eval_rewrite_as_erfczerf._eval_rewrite_as_erfccKt tt|Srr erfirr4r4r5_eval_rewrite_as_erfizerf._eval_rewrite_as_erficCsr|jdj|||d}||d}|tjur$|j|d|dkr dndd}||jvr4|jr4d|tt S| |S)Nrlogxcdirr-+dirr() r)as_leading_termsubsrComplexInfinityr free_symbolsrLrr r.r/r2rrrTarg0r4r4r5_eval_as_leading_terms   zerf._eval_as_leading_termc sddlm}|d}|tjtjfvra|jdz |\}}Wn ttfy-|YSw| }|j rat ||} fddt | D|d| |g} tj t d ttt| Stt|||||S)NrOrdercs>g|]}tj|td|dd|dd|qSr(r8)rrKr.0r]ror4r5 s6z%erf._eval_aseries..r8r()sympy.series.orderrrrHrJr)Zleadterm ValueErrorNotImplementedError is_positiverrangerIrrr rsuperr7 _eval_aseries) r/r[args0r2rrpoint_exZnewns __class__rr5rs&    $zerf._eval_aseriesr8r)!__name__ __module__ __qualname____doc__ unbranchedr=rC classmethodrV staticmethodrr^rdrirlrprrrtrvrrrrrrrrrrrr6r- __classcell__r4r4rr5r71s:L   !    r7c@seZdZdZdZd)ddZd)ddZedd Ze e d d Z d d Z ddZ d*ddZddZddZddZddZddZddZdd Zd!d"Zd#d$Zd%d&ZeZd'd(ZdS)+ra& Complementary Error Function. Explanation =========== The function is defined as: .. math :: \mathrm{erfc}(x) = \frac{2}{\sqrt{\pi}} \int_x^\infty e^{-t^2} \mathrm{d}t Examples ======== >>> from sympy import I, oo, erfc >>> from sympy.abc import z Several special values are known: >>> erfc(0) 1 >>> erfc(oo) 0 >>> erfc(-oo) 2 >>> erfc(I*oo) -oo*I >>> erfc(-I*oo) oo*I The error function obeys the mirror symmetry: >>> from sympy import conjugate >>> conjugate(erfc(z)) erfc(conjugate(z)) Differentiation with respect to $z$ is supported: >>> from sympy import diff >>> diff(erfc(z), z) -2*exp(-z**2)/sqrt(pi) It also follows >>> erfc(-z) 2 - erfc(z) We can numerically evaluate the complementary error function to arbitrary precision on the whole complex plane: >>> erfc(4).evalf(30) 0.0000000154172579002800188521596734869 >>> erfc(4*I).evalf(30) 1.0 - 1296959.73071763923152794095062*I See Also ======== erf: Gaussian error function. erfi: Imaginary error function. erf2: Two-argument error function. erfinv: Inverse error function. erfcinv: Inverse Complementary error function. erf2inv: Inverse two-argument error function. References ========== .. [1] https://en.wikipedia.org/wiki/Error_function .. [2] https://dlmf.nist.gov/7 .. [3] https://mathworld.wolfram.com/Erfc.html .. [4] https://functions.wolfram.com/GammaBetaErf/Erfc Tr8cCs2|dkrdt|jdd ttSt||)Nr8rXrr(r:r;r4r4r5r=or>z erfc.fdiffcCr?r@rNr;r4r4r5rCurDz erfc.inversecCs|jr|tjur tjS|tjurtjS|jrtjSt|tr&tj|j dSt|t r0|j dS|jr6tjS| t }|tjtj fvrF| S|rQd|| SdSNrr()rFrrGrHr,rLrIrMrBr)rNrPr rJrQrRr4r4r5rV|s&      z erfc.evalcGs|dkrtjS|dks|ddkrtjSt|}t|dtd}t|dkr9|d |d|d||Sdtj||||t|tt SrW) rrIr,rrrYrKrrr rZr4r4r5r^s "*zerfc.taylor_termcCr_r`rarcr4r4r5rdrezerfc._eval_conjugatecCs,|jdjdur dS|jdjdurdSdS)NrTF)r)r*rkrcr4r4r5ris zerfc._eval_is_realNcK|tjdd|dSN tractableT)r0rrewriter7r/rorr~r4r4r5rzerfc._eval_rewrite_as_tractablecKrr)rrIr7rr4r4r5_eval_rewrite_as_erfrzerfc._eval_rewrite_as_erfcKstjttt|Sr)rrIr rrr4r4r5rrzerfc._eval_rewrite_as_erficK:tjt|tt}tjtjtt|tt|Srrrr4r4r5r$zerfc._eval_rewrite_as_fresnelscKrrrrr4r4r5rrzerfc._eval_rewrite_as_fresnelcc Ks4tj|ttttjggdgtddg|dSr)rrIrr r&r|r rr4r4r5r4zerfc._eval_rewrite_as_meijergcKs4tjd|ttttjgdtjg|d Sr)rrIrr r%r|rr4r4r5rrzerfc._eval_rewrite_as_hypercKs@ddlm}tjt|d|tj|tj|dttSrw)r{ryrrIrr|r r}r4r4r5rs 4z erfc._eval_rewrite_as_uppergammacKs2tjt|d||ttj|dttSr)rrIrrr|r rr4r4r5rs2zerfc._eval_rewrite_as_expintcK |tSrrr/r1r4r4r5_eval_expand_func zerfc._eval_expand_funccCs^|jdj|||d}||d}|tjur$|j|d|dkr dndd}|jr*tjS||S)Nrrrrrr) r)rrrrrrLrIr.rr4r4r5rs   zerfc._eval_as_leading_termcCstjt|j||||Sr)rrIr7r)r)r/r[rr2rr4r4r5rzerfc._eval_aseriesrr)rrrrrr=rCrrVrrr^rdrirrrrrrrrrrrr6r-rr4r4r4r5r s2L       rcseZdZdZdZd)ddZeddZee dd Z d d Z d d Z ddZ d*ddZddZddZddZddZddZddZdd Zd!d"Zd#d$ZeZd%d&Zfd'd(ZZS)+ra Imaginary error function. Explanation =========== The function erfi is defined as: .. math :: \mathrm{erfi}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{t^2} \mathrm{d}t Examples ======== >>> from sympy import I, oo, erfi >>> from sympy.abc import z Several special values are known: >>> erfi(0) 0 >>> erfi(oo) oo >>> erfi(-oo) -oo >>> erfi(I*oo) I >>> erfi(-I*oo) -I In general one can pull out factors of -1 and $I$ from the argument: >>> erfi(-z) -erfi(z) >>> from sympy import conjugate >>> conjugate(erfi(z)) erfi(conjugate(z)) Differentiation with respect to $z$ is supported: >>> from sympy import diff >>> diff(erfi(z), z) 2*exp(z**2)/sqrt(pi) We can numerically evaluate the imaginary error function to arbitrary precision on the whole complex plane: >>> erfi(2).evalf(30) 18.5648024145755525987042919132 >>> erfi(-2*I).evalf(30) -0.995322265018952734162069256367*I See Also ======== erf: Gaussian error function. erfc: Complementary error function. erf2: Two-argument error function. erfinv: Inverse error function. erfcinv: Inverse Complementary error function. erf2inv: Inverse two-argument error function. References ========== .. [1] https://en.wikipedia.org/wiki/Error_function .. [2] https://mathworld.wolfram.com/Erfi.html .. [3] https://functions.wolfram.com/GammaBetaErf/Erfi Tr8cCs0|dkrdt|jddttSt||r9r:r;r4r4r5r=. z erfi.fdiffcCs|jr|tjur tjS|jrtjS|tjurtjS|jrtjS|r)||  S|t}|durf|tjur9tSt |t rEt|j dSt |t rTttj |j dSt |trh|j djrjt|j dSdSdSdSrE)rFrrGrLr,rHrQrPr rMrBr)rNrIrOrSronzr4r4r5rV4s.       z erfi.evalcGs|dks |ddkr tjSt|}t|dtd}t|dkr1|d|d|d||Sd|||t|ttSrW)rr,rrrYrrr rZr4r4r5r^Rs   zerfi.taylor_termcCr_r`rarcr4r4r5rd_rezerfi._eval_conjugatecC |jdjSr`rhrcr4r4r5_eval_is_extended_realb zerfi._eval_is_extended_realcCrr`r)rLrcr4r4r5rrerzerfi._eval_is_zeroNcKrrrrr4r4r5rhrzerfi._eval_rewrite_as_tractablecKrr)r r7rr4r4r5rkrzerfi._eval_rewrite_as_erfcKsttt|tSr)r rrr4r4r5rnrezerfi._eval_rewrite_as_erfccK4tjt|tt}tjtt|tt|Srrrr4r4r5rqrzerfi._eval_rewrite_as_fresnelscKrrrrr4r4r5rurzerfi._eval_rewrite_as_fresnelccKs0|ttttjggdgtddg|d Srrrr4r4r5ry0zerfi._eval_rewrite_as_meijergcKs,d|ttttjgdtjg|dSrrrr4r4r5r|rzerfi._eval_rewrite_as_hypercKs>ddlm}t|d ||tj|d tttjSrw)r{ryrrr|r rIr}r4r4r5rs 2z erfi._eval_rewrite_as_uppergammacKs0t|d ||ttj|d ttSrrrr4r4r5rrzerfi._eval_rewrite_as_expintcKrrrrr4r4r5rrzerfi._eval_expand_funccCs\|jdj|||d}||d}||jvr!|jr!d|ttS|jr)||S||S)Nrrr() r)rrrrLrr rmr.rr4r4r5rs   zerfi._eval_as_leading_termcsddlm}|d}|tjur:|jdfddt|D|d||g}t tdtt t |St t | ||||S)Nrrcs4g|]}td|dd|d|dqSr)rrrr4r5rs,z&erfi._eval_aseries..r8r()rrrrHr)rr rrr rrrrr/r[rr2rrrrrrr5rs    "zerfi._eval_aseriesrr)rrrrrr=rrVrrr^rdrrrrrrrrrrrrrr6r-rrrr4r4rr5rs2I      rc@seZdZdZddZeddZddZdd Zd d Z d d Z ddZ ddZ ddZ ddZddZddZddZddZddZd S)!erf2a? Two-argument error function. Explanation =========== This function is defined as: .. math :: \mathrm{erf2}(x, y) = \frac{2}{\sqrt{\pi}} \int_x^y e^{-t^2} \mathrm{d}t Examples ======== >>> from sympy import oo, erf2 >>> from sympy.abc import x, y Several special values are known: >>> erf2(0, 0) 0 >>> erf2(x, x) 0 >>> erf2(x, oo) 1 - erf(x) >>> erf2(x, -oo) -erf(x) - 1 >>> erf2(oo, y) erf(y) - 1 >>> erf2(-oo, y) erf(y) + 1 In general one can pull out factors of -1: >>> erf2(-x, -y) -erf2(x, y) The error function obeys the mirror symmetry: >>> from sympy import conjugate >>> conjugate(erf2(x, y)) erf2(conjugate(x), conjugate(y)) Differentiation with respect to $x$, $y$ is supported: >>> from sympy import diff >>> diff(erf2(x, y), x) -2*exp(-x**2)/sqrt(pi) >>> diff(erf2(x, y), y) 2*exp(-y**2)/sqrt(pi) See Also ======== erf: Gaussian error function. erfc: Complementary error function. erfi: Imaginary error function. erfinv: Inverse error function. erfcinv: Inverse Complementary error function. erf2inv: Inverse two-argument error function. References ========== .. [1] https://functions.wolfram.com/GammaBetaErf/Erf2/ cCsX|j\}}|dkrdt|d ttS|dkr'dt|d ttSt||)Nr8rXr()r)rrr rr/r<r2r3r4r4r5r=s  z erf2.fdiffcCstjtjtjf}|tjus|tjurtjS||krtjS||vs$||vr,t|t|St|tr=|jd|kr=|jdS|j sO|j sO|j rI|j sO|j rW|j rWt|t|S| }| }|rk|rk|| |  S|so|rwt|t|SdSrE) rrHrJr,rGr7rMrOr)rLr* is_infiniterQ)rSr2r3ZchkZsign_xZsign_yr4r4r5rVs, z erf2.evalcCs ||jd|jdSrErarcr4r4r5rd s zerf2._eval_conjugatecCs|jdjo |jdjSrErhrcr4r4r5r zerf2._eval_is_extended_realcKst|t|Srr7r/r2r3r~r4r4r5rzerf2._eval_rewrite_as_erfcKst|t|Srrrr4r4r5rrzerf2._eval_rewrite_as_erfccKsttt|tt|Srrrr4r4r5rrzerf2._eval_rewrite_as_erficKt|tt|tSr)r7rrrr4r4r5rrzerf2._eval_rewrite_as_fresnelscKrr)r7rrrr4r4r5rrzerf2._eval_rewrite_as_fresnelccKrr)r7rr&rr4r4r5rrzerf2._eval_rewrite_as_meijergcKrr)r7rr%rr4r4r5r"rzerf2._eval_rewrite_as_hypercKshddlm}t|d|tj|tj|dttt|d|tj|tj|dttSrwrz)r/r2r3r~ryr4r4r5r%s ,,z erf2._eval_rewrite_as_uppergammacKrr)r7rrrr4r4r5r*rzerf2._eval_rewrite_as_expintcKrrrrr4r4r5r-rzerf2._eval_expand_funccCs t|jSr)r r)rcr4r4r5rr0rzerf2._eval_is_zeroN)rrrrr=rrVrdrrrrrrrrrrrrrr4r4r4r5rs$E  rc@s@eZdZdZdddZdddZeddZd d Zd d Z d S)rBaR Inverse Error Function. The erfinv function is defined as: .. math :: \mathrm{erf}(x) = y \quad \Rightarrow \quad \mathrm{erfinv}(y) = x Examples ======== >>> from sympy import erfinv >>> from sympy.abc import x Several special values are known: >>> erfinv(0) 0 >>> erfinv(1) oo Differentiation with respect to $x$ is supported: >>> from sympy import diff >>> diff(erfinv(x), x) sqrt(pi)*exp(erfinv(x)**2)/2 We can numerically evaluate the inverse error function to arbitrary precision on [-1, 1]: >>> erfinv(0.2).evalf(30) 0.179143454621291692285822705344 See Also ======== erf: Gaussian error function. erfc: Complementary error function. erfi: Imaginary error function. erf2: Two-argument error function. erfcinv: Inverse Complementary error function. erf2inv: Inverse two-argument error function. References ========== .. [1] https://en.wikipedia.org/wiki/Error_function#Inverse_functions .. [2] https://functions.wolfram.com/GammaBetaErf/InverseErf/ r8cCs8|dkrttt||jddtjSt||Nr8rr(rr rr.r)rr|rr;r4r4r5r=fs& z erfinv.fdiffcCr?r@rr;r4r4r5rClrDzerfinv.inversecCs|tjurtjS|tjurtjS|jrtjS|tjurtjSt|t r.|j dj r.|j dS|jr4tjS| d}|durNt|t rP|j dj rR|j d SdSdSdSNrr) rrGrKrJrLr,rIrHrMr7r)r*rPrr4r4r5rVss       z erfinv.evalcK td|SNr8rrr4r4r5_eval_rewrite_as_erfcinvrzerfinv._eval_rewrite_as_erfcinvcCrr`rrcr4r4r5rrrzerfinv._eval_is_zeroNr) rrrrr=rCrrVrrrr4r4r4r5rB3s 2   rBc@sHeZdZdZdddZdddZeddZd d Zd d Z d dZ dS)rNa Inverse Complementary Error Function. The erfcinv function is defined as: .. math :: \mathrm{erfc}(x) = y \quad \Rightarrow \quad \mathrm{erfcinv}(y) = x Examples ======== >>> from sympy import erfcinv >>> from sympy.abc import x Several special values are known: >>> erfcinv(1) 0 >>> erfcinv(0) oo Differentiation with respect to $x$ is supported: >>> from sympy import diff >>> diff(erfcinv(x), x) -sqrt(pi)*exp(erfcinv(x)**2)/2 See Also ======== erf: Gaussian error function. erfc: Complementary error function. erfi: Imaginary error function. erf2: Two-argument error function. erfinv: Inverse error function. erf2inv: Inverse two-argument error function. References ========== .. [1] https://en.wikipedia.org/wiki/Error_function#Inverse_functions .. [2] https://functions.wolfram.com/GammaBetaErf/InverseErfc/ r8cCs:|dkrtt t||jddtjSt||rrr;r4r4r5r=s( z erfcinv.fdiffcCr?r@rr;r4r4r5rCrDzerfcinv.inversecCsJ|tjurtjS|jrtjS|tjurtjS|dkrtjS|jr#tjSdSr)rrGrLrHrIr,rJrSror4r4r5rVs  z erfcinv.evalcKrrrArr4r4r5_eval_rewrite_as_erfinvrzerfcinv._eval_rewrite_as_erfinvcCs|jddjSrErrcr4r4r5rrrzerfcinv._eval_is_zerocCs"|jd}t|jt|tdgSr)r)rrLr r rnr4r4r5_eval_is_infinites zerfcinv._eval_is_infiniteNr) rrrrr=rCrrVrrrrr4r4r4r5rNs ,    rNc@s,eZdZdZddZeddZddZdS) rOa2 Two-argument Inverse error function. The erf2inv function is defined as: .. math :: \mathrm{erf2}(x, w) = y \quad \Rightarrow \quad \mathrm{erf2inv}(x, y) = w Examples ======== >>> from sympy import erf2inv, oo >>> from sympy.abc import x, y Several special values are known: >>> erf2inv(0, 0) 0 >>> erf2inv(1, 0) 1 >>> erf2inv(0, 1) oo >>> erf2inv(0, y) erfinv(y) >>> erf2inv(oo, y) erfcinv(-y) Differentiation with respect to $x$ and $y$ is supported: >>> from sympy import diff >>> diff(erf2inv(x, y), x) exp(-x**2 + erf2inv(x, y)**2) >>> diff(erf2inv(x, y), y) sqrt(pi)*exp(erf2inv(x, y)**2)/2 See Also ======== erf: Gaussian error function. erfc: Complementary error function. erfi: Imaginary error function. erf2: Two-argument error function. erfinv: Inverse error function. erfcinv: Inverse complementary error function. References ========== .. [1] https://functions.wolfram.com/GammaBetaErf/InverseErf2/ cCsb|j\}}|dkrt|||d|dS|dkr,tttjt|||dSt||)Nr8r()r)rr.rr rr|rrr4r4r5r=s " z erf2inv.fdiffcCs|tjus |tjur tjS|jr|jrtjS|jr!|tjur!tjS|tjur,|jr,tjS|jr3t|S|tjur=t| S|jrB|S|tjurKt|S|jrX|jrTtjSt|S|jr]|SdSr)rrGrLr,rIrHrBrN)rSr2r3r4r4r5rV s.    z erf2inv.evalcCs"|j\}}|jr |jrdSdSdS)NTr)r/r2r3r4r4r5rr;s  zerf2inv._eval_is_zeroN)rrrrr=rrVrrr4r4r4r5rOs 3  rOcseZdZdZeddZdddZfddZd d Zd d Z d dZ ddZ e Z e Z e ZdddZddZfddZdfdd ZfddZZS) Eia The classical exponential integral. Explanation =========== For use in SymPy, this function is defined as .. math:: \operatorname{Ei}(x) = \sum_{n=1}^\infty \frac{x^n}{n\, n!} + \log(x) + \gamma, where $\gamma$ is the Euler-Mascheroni constant. If $x$ is a polar number, this defines an analytic function on the Riemann surface of the logarithm. Otherwise this defines an analytic function in the cut plane $\mathbb{C} \setminus (-\infty, 0]$. **Background** The name exponential integral comes from the following statement: .. math:: \operatorname{Ei}(x) = \int_{-\infty}^x \frac{e^t}{t} \mathrm{d}t If the integral is interpreted as a Cauchy principal value, this statement holds for $x > 0$ and $\operatorname{Ei}(x)$ as defined above. Examples ======== >>> from sympy import Ei, polar_lift, exp_polar, I, pi >>> from sympy.abc import x >>> Ei(-1) Ei(-1) This yields a real value: >>> Ei(-1).n(chop=True) -0.219383934395520 On the other hand the analytic continuation is not real: >>> Ei(polar_lift(-1)).n(chop=True) -0.21938393439552 + 3.14159265358979*I The exponential integral has a logarithmic branch point at the origin: >>> Ei(x*exp_polar(2*I*pi)) Ei(x) + 2*I*pi Differentiation is supported: >>> Ei(x).diff(x) exp(x)/x The exponential integral is related to many other special functions. For example: >>> from sympy import expint, Shi >>> Ei(x).rewrite(expint) -expint(1, x*exp_polar(I*pi)) - I*pi >>> Ei(x).rewrite(Shi) Chi(x) + Shi(x) See Also ======== expint: Generalised exponential integral. E1: Special case of the generalised exponential integral. li: Logarithmic integral. Li: Offset logarithmic integral. Si: Sine integral. Ci: Cosine integral. Shi: Hyperbolic sine integral. Chi: Hyperbolic cosine integral. uppergamma: Upper incomplete gamma function. References ========== .. [1] https://dlmf.nist.gov/6.6 .. [2] https://en.wikipedia.org/wiki/Exponential_integral .. [3] Abramowitz & Stegun, section 5: https://web.archive.org/web/20201128173312/http://people.math.sfu.ca/~cbm/aands/page_228.htm cCsd|jrtjS|tjurtjS|tjurtjS|jrtjS|\}}|r0t|dtt|SdSr) rLrrJrHr,extract_branch_factorrr r rSrorr[r4r4r5rVs   zEi.evalr8cCs,t|jd}|dkrt||St||rE)rr)rrr/r<rTr4r4r5r=s  zEi.fdiffcs:|jdtdjrt|tt|St|Sr)r)rrr _eval_evalfr r r/precrr4r5rs zEi._eval_evalfcKs(ddlm}|dtd| ttS)Nrrxr)r{ryrr r r}r4r4r5rs zEi._eval_rewrite_as_uppergammacKstdtd| ttS)Nr8r)rrr r rr4r4r5rrzEi._eval_rewrite_as_expintcKs$t|tr t|jdStt|Sr`)rMrlir)rrr4r4r5_eval_rewrite_as_lis  zEi._eval_rewrite_as_licKs.|jrt|t|ttSt|t|Sr) is_negativeShiChir r rr4r4r5_eval_rewrite_as_SiszEi._eval_rewrite_as_SiNcKst|t|Sr)r_eisrr4r4r5rrzEi._eval_rewrite_as_tractablecKs:ddlm}ttd|gj}|tj|||tj|fSNrIntegralrU)sympy.integrals.integralsrrrnamerZExp1rJr/ror~rrUr4r4r5_eval_rewrite_as_Integrals zEi._eval_rewrite_as_Integralc sddlm}|jd|d}|jdj||d}|||}|jrJ||\}}|dur1t|n|}t|||t ||j rFt t St jStj|||dS)Nr)r)rr)Zsympyrr)rrrrLas_coeff_exponentrrrr r rr,rr) r/r2rrrx0rTcerr4r5rs  zEi._eval_as_leading_termrcB|jd|d}|jr|j|j}||||St|||Sr`)r)rrLr  _eval_nseriesrr/r2r[rrrfrr4r5r  zEi._eval_nseriescsddlm}|d}|tjtjfvr6|jdfddt|D|d||g}tt|St t | ||||S)Nrrcsg|] }t||qSr4rrrr4r5rz$Ei._eval_aseries..r8) rrrrHrJr)rrrrrrrrrr5rs  zEi._eval_aseriesrrr)rrrrrrVr=rrrrr _eval_rewrite_as_Ci_eval_rewrite_as_Chi_eval_rewrite_as_Shirrrrrrr4r4rr5rDs$W       rcsveZdZdZeddZddZddZdd Zd d Z d d Z e Z e Z e Z dfdd ZfddZddZZS)ra Generalized exponential integral. Explanation =========== This function is defined as .. math:: \operatorname{E}_\nu(z) = z^{\nu - 1} \Gamma(1 - \nu, z), where $\Gamma(1 - \nu, z)$ is the upper incomplete gamma function (``uppergamma``). Hence for $z$ with positive real part we have .. math:: \operatorname{E}_\nu(z) = \int_1^\infty \frac{e^{-zt}}{t^\nu} \mathrm{d}t, which explains the name. The representation as an incomplete gamma function provides an analytic continuation for $\operatorname{E}_\nu(z)$. If $\nu$ is a non-positive integer, the exponential integral is thus an unbranched function of $z$, otherwise there is a branch point at the origin. Refer to the incomplete gamma function documentation for details of the branching behavior. Examples ======== >>> from sympy import expint, S >>> from sympy.abc import nu, z Differentiation is supported. Differentiation with respect to $z$ further explains the name: for integral orders, the exponential integral is an iterated integral of the exponential function. >>> expint(nu, z).diff(z) -expint(nu - 1, z) Differentiation with respect to $\nu$ has no classical expression: >>> expint(nu, z).diff(nu) -z**(nu - 1)*meijerg(((), (1, 1)), ((0, 0, 1 - nu), ()), z) At non-postive integer orders, the exponential integral reduces to the exponential function: >>> expint(0, z) exp(-z)/z >>> expint(-1, z) exp(-z)/z + exp(-z)/z**2 At half-integers it reduces to error functions: >>> expint(S(1)/2, z) sqrt(pi)*erfc(sqrt(z))/sqrt(z) At positive integer orders it can be rewritten in terms of exponentials and ``expint(1, z)``. Use ``expand_func()`` to do this: >>> from sympy import expand_func >>> expand_func(expint(5, z)) z**4*expint(1, z)/24 + (-z**3 + z**2 - 2*z + 6)*exp(-z)/24 The generalised exponential integral is essentially equivalent to the incomplete gamma function: >>> from sympy import uppergamma >>> expint(nu, z).rewrite(uppergamma) z**(nu - 1)*uppergamma(1 - nu, z) As such it is branched at the origin: >>> from sympy import exp_polar, pi, I >>> expint(4, z*exp_polar(2*pi*I)) I*pi*z**3/3 + expint(4, z) >>> expint(nu, z*exp_polar(2*pi*I)) z**(nu - 1)*(exp(2*I*pi*nu) - 1)*gamma(1 - nu) + expint(nu, z) See Also ======== Ei: Another related function called exponential integral. E1: The classical case, returns expint(1, z). li: Logarithmic integral. Li: Offset logarithmic integral. Si: Sine integral. Ci: Cosine integral. Shi: Hyperbolic sine integral. Chi: Hyperbolic cosine integral. uppergamma References ========== .. [1] https://dlmf.nist.gov/8.19 .. [2] https://functions.wolfram.com/GammaBetaErf/ExpIntegralE/ .. [3] https://en.wikipedia.org/wiki/Exponential_integral cCsddlm}m}t|}||krt||S|jr|dks$|js5d|jr5tt||d|d||S|\}}|tj urBdS|j rm|dksKdSt||dt t |tj |dt|dt||dStdt t ||d||d|d|t||S)Nr)gammaryr(r8)r{r#ryrr is_Integerrrrr, is_integerr r rKrr)rSnuror#ryZnu2r[r4r4r5rVcs  "  8>z expint.evalcCs`|j\}}|dkr||d tgddgddd|gg|S|dkr+t|d| St||r)r)r&rr)r/r<r&ror4r4r5r=ys , z expint.fdiffcKs&ddlm}||d|d||S)Nrrxr8)r{ry)r/r&ror~ryr4r4r5rs z"expint._eval_rewrite_as_uppergammac sdkrt|tt t ttSjrMdkrMt| dtdt|tt tdt fddt dDS|S)Nr8cs$g|]}t|d|qSr(rrr&r2r4r5r$z.expint._eval_rewrite_as_Ei..) rrr r r$rrE1rrrrr/r&ror~r4r(r5_eval_rewrite_as_Eis  $zexpint._eval_rewrite_as_EicKs|tjtfi|Sr)rrrrr4r4r5rrzexpint._eval_expand_funccKs|dkr|St|t|Sr)r r r+r4r4r5r szexpint._eval_rewrite_as_Sircst|jd|s2|jd}|dkr|j|j}||||S|jr2|dkr2|j|j}||||St|||SrE)r)hasr rr$r,r)r/r2r[rrr&rrr4r5rs   zexpint._eval_nseriescsddlm}|d}|jd|tjur:|jdfddt|D|d||g}t t|Stt | ||||S)Nrrr8cs(g|]}tj|t||qSr4)rrKrrr&ror4r5rs(z(expint._eval_aseries..) rrr)rrHrrrrrrrrr.r5rs    ,zexpint._eval_aseriescOsLddlm}|j\}}ttd|j}||| t| ||dtjfSNrrrUr8) rrr)rrrrrrH)r/r)r~rr[r2rUr4r4r5rs  &z expint._eval_rewrite_as_Integralr)rrrrrrVr=rr,rr r r!r"rrrrr4r4rr5rsg     rcCs td|S)a+ Classical case of the generalized exponential integral. Explanation =========== This is equivalent to ``expint(1, z)``. Examples ======== >>> from sympy import E1 >>> E1(0) expint(1, 0) >>> E1(5) expint(1, 5) See Also ======== Ei: Exponential integral. expint: Generalised exponential integral. li: Logarithmic integral. Li: Offset logarithmic integral. Si: Sine integral. Ci: Cosine integral. Shi: Hyperbolic sine integral. Chi: Hyperbolic cosine integral. r8)rrr4r4r5r*s r*c@seZdZdZeddZdddZddZd d Zd d Z d dZ ddZ e Z ddZ e ZddZddZd ddZd!ddZddZdS)"ra The classical logarithmic integral. Explanation =========== For use in SymPy, this function is defined as .. math:: \operatorname{li}(x) = \int_0^x \frac{1}{\log(t)} \mathrm{d}t \,. Examples ======== >>> from sympy import I, oo, li >>> from sympy.abc import z Several special values are known: >>> li(0) 0 >>> li(1) -oo >>> li(oo) oo Differentiation with respect to $z$ is supported: >>> from sympy import diff >>> diff(li(z), z) 1/log(z) Defining the ``li`` function via an integral: >>> from sympy import integrate >>> integrate(li(z)) z*li(z) - Ei(2*log(z)) >>> integrate(li(z),z) z*li(z) - Ei(2*log(z)) The logarithmic integral can also be defined in terms of ``Ei``: >>> from sympy import Ei >>> li(z).rewrite(Ei) Ei(log(z)) >>> diff(li(z).rewrite(Ei), z) 1/log(z) We can numerically evaluate the logarithmic integral to arbitrary precision on the whole complex plane (except the singular points): >>> li(2).evalf(30) 1.04516378011749278484458888919 >>> li(2*I).evalf(30) 1.0652795784357498247001125598 + 3.08346052231061726610939702133*I We can even compute Soldner's constant by the help of mpmath: >>> from mpmath import findroot >>> findroot(li, 2) 1.45136923488338 Further transformations include rewriting ``li`` in terms of the trigonometric integrals ``Si``, ``Ci``, ``Shi`` and ``Chi``: >>> from sympy import Si, Ci, Shi, Chi >>> li(z).rewrite(Si) -log(I*log(z)) - log(1/log(z))/2 + log(log(z))/2 + Ci(I*log(z)) + Shi(log(z)) >>> li(z).rewrite(Ci) -log(I*log(z)) - log(1/log(z))/2 + log(log(z))/2 + Ci(I*log(z)) + Shi(log(z)) >>> li(z).rewrite(Shi) -log(1/log(z))/2 + log(log(z))/2 + Chi(log(z)) - Shi(log(z)) >>> li(z).rewrite(Chi) -log(1/log(z))/2 + log(log(z))/2 + Chi(log(z)) - Shi(log(z)) See Also ======== Li: Offset logarithmic integral. Ei: Exponential integral. expint: Generalised exponential integral. E1: Special case of the generalised exponential integral. Si: Sine integral. Ci: Cosine integral. Shi: Hyperbolic sine integral. Chi: Hyperbolic cosine integral. References ========== .. [1] https://en.wikipedia.org/wiki/Logarithmic_integral .. [2] https://mathworld.wolfram.com/LogarithmicIntegral.html .. [3] https://dlmf.nist.gov/6 .. [4] https://mathworld.wolfram.com/SoldnersConstant.html cCs<|jrtjS|tjurtjS|tjurtjS|jrtjSdSr)rLrr,rIrJrHrr4r4r5rVBs  zli.evalr8cC*|jd}|dkrtjt|St||rEr)rrIrrrr4r4r5r=M  zli.fdiffcCs"|jd}|js||SdSr`)r)rur.rbrnr4r4r5rdTs zli._eval_conjugatecKst|tdSr)Lirrr4r4r5_eval_rewrite_as_LiZrzli._eval_rewrite_as_LicKs tt|Sr)rrrr4r4r5r,]rzli._eval_rewrite_as_EicKsPddlm}|dt|  tjtt|ttjt|tt| SNrrx)r{ryrrr|rIr}r4r4r5r`s " zli._eval_rewrite_as_uppergammacKsXttt|tttt|tjttjt|tt|ttt|Sr)Cir rSirr|rIrr4r4r5r es ""zli._eval_rewrite_as_SicKs<tt|tt|tjttjt|tt|Sr)r rr rr|rIrr4r4r5r"ks<zli._eval_rewrite_as_ShicKs@t|tddt|tjtt|ttjt|tS)N)r8r8)r(r()rr%rr|rIrrr4r4r5rps "zli._eval_rewrite_as_hypercKsFtt|  tjttjt|tt|tddt| S)N)r4r))rrr4)rrr|rIr&rr4r4r5rts2zli._eval_rewrite_as_meijergNcKs|tt|Sr)r rrr4r4r5rxrzli._eval_rewrite_as_tractablercs:|jdfddtd|D}tttt|S)Nrcs$g|]}t|t||qSr4)rrrrr4r5r}r)z$li._eval_nseries..r8)r)rrrr)r/r2r[rrrr4rr5r{s zli._eval_nseriescC|jd}|jr dSdSrgrrnr4r4r5rr zli._eval_is_zerorrr)rrrrrrVr=rdr4r,rr r r"r!rrrrrrr4r4r4r5rs$c     rc@sJeZdZdZeddZdddZddZd d Zdd d Z dddZ d S)r3ab The offset logarithmic integral. Explanation =========== For use in SymPy, this function is defined as .. math:: \operatorname{Li}(x) = \operatorname{li}(x) - \operatorname{li}(2) Examples ======== >>> from sympy import Li >>> from sympy.abc import z The following special value is known: >>> Li(2) 0 Differentiation with respect to $z$ is supported: >>> from sympy import diff >>> diff(Li(z), z) 1/log(z) The shifted logarithmic integral can be written in terms of $li(z)$: >>> from sympy import li >>> Li(z).rewrite(li) li(z) - li(2) We can numerically evaluate the logarithmic integral to arbitrary precision on the whole complex plane (except the singular points): >>> Li(2).evalf(30) 0 >>> Li(4).evalf(30) 1.92242131492155809316615998938 See Also ======== li: Logarithmic integral. Ei: Exponential integral. expint: Generalised exponential integral. E1: Special case of the generalised exponential integral. Si: Sine integral. Ci: Cosine integral. Shi: Hyperbolic sine integral. Chi: Hyperbolic cosine integral. References ========== .. [1] https://en.wikipedia.org/wiki/Logarithmic_integral .. [2] https://mathworld.wolfram.com/LogarithmicIntegral.html .. [3] https://dlmf.nist.gov/6 cCs&|tjurtjS|tdkrtjSdSr)rrHr,rr4r4r5rVs  zLi.evalr8cCr0rEr1rr4r4r5r=r2zLi.fdiffcCs|t|Sr)rrZevalfrr4r4r5rrzLi._eval_evalfcKst|tdSr)rrr4r4r5rrzLi._eval_rewrite_as_liNcKs|tjdddS)NrT)r0)rrrr4r4r5rrezLi._eval_rewrite_as_tractablercCs|j|j}||||Sr)rr)r)r/r2r[rrrr4r4r5rs zLi._eval_nseriesrrr) rrrrrrVr=rrrrr4r4r4r5r3s@   r3csHeZdZdZeddZdddZddZd d Zdfd d Z Z S)TrigonometricIntegralz) Base class for trigonometric integrals. cCs(|tjur|jS|tjur|S|tjur|S|jr |jS|t t }|dur7| ddkr7|t }|durA| |dS|t t }|durS| |dS|t d}|durj| ddkrj|d}|durs| |S|\}}|dkr||krdSdtt || d||S)Nrr8rr()rr,_atzerorH_atinfrJ _atneginfrLrPrr _trigfunc_Ifactor _minusfactorrr rr4r4r5rVs2         "zTrigonometricIntegral.evalr8cCs.t|jd}|dkr|||St||rE)rr)r>rrr4r4r5r= s zTrigonometricIntegral.fdiffcKs||tSr)rrrrr4r4r5r,rz)TrigonometricIntegral._eval_rewrite_as_EicKsddlm}|||Sr5)r{ryrrr}r4r4r5rs z1TrigonometricIntegral._eval_rewrite_as_uppergammarcs|jd|ddkrt|||S|||||}|ddkr(|d8}|jtdddd}|ddkrA|tt|7}|||jd|||S)Nrr8cSs |||Srr4)rUr[r4r4r5s z5TrigonometricIntegral._eval_nseries..F)Z simultaneous) r)rrrr>replacerrr)r/r2r[rrZ baseseriesrr4r5rsz#TrigonometricIntegral._eval_nseriesrr) rrrrrrVr=r,rrrr4r4rr5r:s  r:cs~eZdZdZeZejZe ddZ e ddZ e ddZ e dd Z d d Zd d ZeZddZfddZddZZS)r7a Sine integral. Explanation =========== This function is defined by .. math:: \operatorname{Si}(z) = \int_0^z \frac{\sin{t}}{t} \mathrm{d}t. It is an entire function. Examples ======== >>> from sympy import Si >>> from sympy.abc import z The sine integral is an antiderivative of $sin(z)/z$: >>> Si(z).diff(z) sin(z)/z It is unbranched: >>> from sympy import exp_polar, I, pi >>> Si(z*exp_polar(2*I*pi)) Si(z) Sine integral behaves much like ordinary sine under multiplication by ``I``: >>> Si(I*z) I*Shi(z) >>> Si(-z) -Si(z) It can also be expressed in terms of exponential integrals, but beware that the latter is branched: >>> from sympy import expint >>> Si(z).rewrite(expint) -I*(-expint(1, z*exp_polar(-I*pi/2))/2 + expint(1, z*exp_polar(I*pi/2))/2) + pi/2 It can be rewritten in the form of sinc function (by definition): >>> from sympy import sinc >>> Si(z).rewrite(sinc) Integral(sinc(_t), (_t, 0, z)) See Also ======== Ci: Cosine integral. Shi: Hyperbolic sine integral. Chi: Hyperbolic cosine integral. Ei: Exponential integral. expint: Generalised exponential integral. sinc: unnormalized sinc function E1: Special case of the generalised exponential integral. li: Logarithmic integral. Li: Offset logarithmic integral. References ========== .. [1] https://en.wikipedia.org/wiki/Trigonometric_integral cCs ttjSrr rr|rSr4r4r5r<n z Si._atinfcCs t tjSrrCrDr4r4r5r=rs z Si._atneginfcC t| Sr)r7rr4r4r5r@vrEzSi._minusfactorcCtt||Sr)r r rSrosignr4r4r5r?zz Si._IfactorcKs2tdttt|ttt |dtSr)r r*rr rr4r4r5r~s2zSi._eval_rewrite_as_expintcKs2ddlm}ttd|gj}|t||d|fSr )rrrrrr$rr4r4r5rs zSi._eval_rewrite_as_IntegralcCsh|jdj|||d}||d}|tjur%|j|dt|jr!dndd}|jr*|S|j s2| |S|SNrrrrr r)rrrrGrrrrLrr.rr4r4r5rs   zSi._eval_as_leading_termc sddlm}|d}|tjurX|jdfddt|ddD|d||g}fddt|dD|d||g}tdtt|t t|St t | ||||S)Nrrc2g|]}tj|td|d|dqSrrrKrrrr4r5r*z$Si._eval_aseries..r(r8c6g|]}tj|td|dd|dqSrrNrrr4r5r.) rrrrHr)rr r"rr#rr7r) r/r[rr2rrrpqrrr5rs      (zSi._eval_aseriescCr8rgrrnr4r4r5rrr9zSi._eval_is_zero)rrrrr#r>rr,r;rr<r=r@r?rrZ_eval_rewrite_as_sincrrrrrr4r4rr5r7$s$F     r7csreZdZdZeZejZe ddZ e ddZ e ddZ e dd Z d d Zd d ZddZfddZZS)r6a Cosine integral. Explanation =========== This function is defined for positive $x$ by .. math:: \operatorname{Ci}(x) = \gamma + \log{x} + \int_0^x \frac{\cos{t} - 1}{t} \mathrm{d}t = -\int_x^\infty \frac{\cos{t}}{t} \mathrm{d}t, where $\gamma$ is the Euler-Mascheroni constant. We have .. math:: \operatorname{Ci}(z) = -\frac{\operatorname{E}_1\left(e^{i\pi/2} z\right) + \operatorname{E}_1\left(e^{-i \pi/2} z\right)}{2} which holds for all polar $z$ and thus provides an analytic continuation to the Riemann surface of the logarithm. The formula also holds as stated for $z \in \mathbb{C}$ with $\Re(z) > 0$. By lifting to the principal branch, we obtain an analytic function on the cut complex plane. Examples ======== >>> from sympy import Ci >>> from sympy.abc import z The cosine integral is a primitive of $\cos(z)/z$: >>> Ci(z).diff(z) cos(z)/z It has a logarithmic branch point at the origin: >>> from sympy import exp_polar, I, pi >>> Ci(z*exp_polar(2*I*pi)) Ci(z) + 2*I*pi The cosine integral behaves somewhat like ordinary $\cos$ under multiplication by $i$: >>> from sympy import polar_lift >>> Ci(polar_lift(I)*z) Chi(z) + I*pi/2 >>> Ci(polar_lift(-1)*z) Ci(z) + I*pi It can also be expressed in terms of exponential integrals: >>> from sympy import expint >>> Ci(z).rewrite(expint) -expint(1, z*exp_polar(-I*pi/2))/2 - expint(1, z*exp_polar(I*pi/2))/2 See Also ======== Si: Sine integral. Shi: Hyperbolic sine integral. Chi: Hyperbolic cosine integral. Ei: Exponential integral. expint: Generalised exponential integral. E1: Special case of the generalised exponential integral. li: Logarithmic integral. Li: Offset logarithmic integral. References ========== .. [1] https://en.wikipedia.org/wiki/Trigonometric_integral cCtjSr)rr,rDr4r4r5r<z Ci._atinfcCsttSr)r r rDr4r4r5r=sz Ci._atneginfcCt|ttSrr6r r rr4r4r5r@rJzCi._minusfactorcCt|ttd|Srr r r rHr4r4r5r? z Ci._IfactorcKs(ttt|ttt | dSr)r*rr rr4r4r5rs(zCi._eval_rewrite_as_expintcKsHddlm}ttd|gj}tjt||dt|||d|fSr/) rrrrrrrrr"rr4r4r5rs *zCi._eval_rewrite_as_IntegralcC|jdj|||d}||d}|tjur%|j|dt|jr!dndd}|jrC| |\}}|dur7t |n|}t |||t S|j rK| |S|SrKr)rrrrGrrrrLrrrrmr.r/r2rrrTrrrr4r4r5r   zCi._eval_as_leading_termc sddlm}|d}|tjtjfvrd|jdfddt|ddD|d||g}fddt|dD|d||g}tt|t t|} |tjurb| t t 7} | St t |||||S)NrrcrMrrNrrr4r5r,rOz$Ci._eval_aseries..r(r8crPrrNrrr4r5r.rQ)rrrrHrJr)rr#rr"r r rr6r) r/r[rr2rrrrRrSresultrrr5r&s&        zCi._eval_aseries)rrrrr"r>rrr;rr<r=r@r?rrrrrr4r4rr5r6s O    r6c@sbeZdZdZeZejZe ddZ e ddZ e ddZ e dd Z d d Zd d ZddZdS)r a Sinh integral. Explanation =========== This function is defined by .. math:: \operatorname{Shi}(z) = \int_0^z \frac{\sinh{t}}{t} \mathrm{d}t. It is an entire function. Examples ======== >>> from sympy import Shi >>> from sympy.abc import z The Sinh integral is a primitive of $\sinh(z)/z$: >>> Shi(z).diff(z) sinh(z)/z It is unbranched: >>> from sympy import exp_polar, I, pi >>> Shi(z*exp_polar(2*I*pi)) Shi(z) The $\sinh$ integral behaves much like ordinary $\sinh$ under multiplication by $i$: >>> Shi(I*z) I*Si(z) >>> Shi(-z) -Shi(z) It can also be expressed in terms of exponential integrals, but beware that the latter is branched: >>> from sympy import expint >>> Shi(z).rewrite(expint) expint(1, z)/2 - expint(1, z*exp_polar(I*pi))/2 - I*pi/2 See Also ======== Si: Sine integral. Ci: Cosine integral. Chi: Hyperbolic cosine integral. Ei: Exponential integral. expint: Generalised exponential integral. E1: Special case of the generalised exponential integral. li: Logarithmic integral. Li: Offset logarithmic integral. References ========== .. [1] https://en.wikipedia.org/wiki/Trigonometric_integral cCrTrrrHrDr4r4r5r<{rUz Shi._atinfcCrTr)rrJrDr4r4r5r=rUz Shi._atneginfcCrFr)r rr4r4r5r@rEzShi._minusfactorcCrGr)r r7rHr4r4r5r?rJz Shi._IfactorcKs,t|tttt|dttdSr)r*rr r rr4r4r5rs,zShi._eval_rewrite_as_expintcCr8rgrrnr4r4r5rrr9zShi._eval_is_zerocCsb|jd|}||d}|tjur"|j|dt|jrdndd}|jr'|S|j s/| |S|S)NrrrrrLrr4r4r5rs   zShi._eval_as_leading_termN)rrrrr!r>rr,r;rr<r=r@r?rrrrr4r4r4r5r 8s?     r c@sZeZdZdZeZejZe ddZ e ddZ e ddZ e dd Z d d Zd d ZdS)r a  Cosh integral. Explanation =========== This function is defined for positive $x$ by .. math:: \operatorname{Chi}(x) = \gamma + \log{x} + \int_0^x \frac{\cosh{t} - 1}{t} \mathrm{d}t, where $\gamma$ is the Euler-Mascheroni constant. We have .. math:: \operatorname{Chi}(z) = \operatorname{Ci}\left(e^{i \pi/2}z\right) - i\frac{\pi}{2}, which holds for all polar $z$ and thus provides an analytic continuation to the Riemann surface of the logarithm. By lifting to the principal branch we obtain an analytic function on the cut complex plane. Examples ======== >>> from sympy import Chi >>> from sympy.abc import z The $\cosh$ integral is a primitive of $\cosh(z)/z$: >>> Chi(z).diff(z) cosh(z)/z It has a logarithmic branch point at the origin: >>> from sympy import exp_polar, I, pi >>> Chi(z*exp_polar(2*I*pi)) Chi(z) + 2*I*pi The $\cosh$ integral behaves somewhat like ordinary $\cosh$ under multiplication by $i$: >>> from sympy import polar_lift >>> Chi(polar_lift(I)*z) Ci(z) + I*pi/2 >>> Chi(polar_lift(-1)*z) Chi(z) + I*pi It can also be expressed in terms of exponential integrals: >>> from sympy import expint >>> Chi(z).rewrite(expint) -expint(1, z)/2 - expint(1, z*exp_polar(I*pi))/2 - I*pi/2 See Also ======== Si: Sine integral. Ci: Cosine integral. Shi: Hyperbolic sine integral. Ei: Exponential integral. expint: Generalised exponential integral. E1: Special case of the generalised exponential integral. li: Logarithmic integral. Li: Offset logarithmic integral. References ========== .. 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Explanation =========== This function is defined by .. math:: \operatorname{S}(z) = \int_0^z \sin{\frac{\pi}{2} t^2} \mathrm{d}t. It is an entire function. Examples ======== >>> from sympy import I, oo, fresnels >>> from sympy.abc import z Several special values are known: >>> fresnels(0) 0 >>> fresnels(oo) 1/2 >>> fresnels(-oo) -1/2 >>> fresnels(I*oo) -I/2 >>> fresnels(-I*oo) I/2 In general one can pull out factors of -1 and $i$ from the argument: >>> fresnels(-z) -fresnels(z) >>> fresnels(I*z) -I*fresnels(z) The Fresnel S integral obeys the mirror symmetry $\overline{S(z)} = S(\bar{z})$: >>> from sympy import conjugate >>> conjugate(fresnels(z)) fresnels(conjugate(z)) Differentiation with respect to $z$ is supported: >>> from sympy import diff >>> diff(fresnels(z), z) sin(pi*z**2/2) Defining the Fresnel functions via an integral: >>> from sympy import integrate, pi, sin, expand_func >>> integrate(sin(pi*z**2/2), z) 3*fresnels(z)*gamma(3/4)/(4*gamma(7/4)) >>> expand_func(integrate(sin(pi*z**2/2), z)) fresnels(z) We can numerically evaluate the Fresnel integral to arbitrary precision on the whole complex plane: >>> fresnels(2).evalf(30) 0.343415678363698242195300815958 >>> fresnels(-2*I).evalf(30) 0.343415678363698242195300815958*I See Also ======== fresnelc: Fresnel cosine integral. References ========== .. [1] https://en.wikipedia.org/wiki/Fresnel_integral .. [2] https://dlmf.nist.gov/7 .. [3] https://mathworld.wolfram.com/FresnelIntegrals.html .. [4] https://functions.wolfram.com/GammaBetaErf/FresnelS .. [5] The converging factors for the fresnel integrals by John W. 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Explanation =========== This function is defined by .. math:: \operatorname{C}(z) = \int_0^z \cos{\frac{\pi}{2} t^2} \mathrm{d}t. It is an entire function. Examples ======== >>> from sympy import I, oo, fresnelc >>> from sympy.abc import z Several special values are known: >>> fresnelc(0) 0 >>> fresnelc(oo) 1/2 >>> fresnelc(-oo) -1/2 >>> fresnelc(I*oo) I/2 >>> fresnelc(-I*oo) -I/2 In general one can pull out factors of -1 and $i$ from the argument: >>> fresnelc(-z) -fresnelc(z) >>> fresnelc(I*z) I*fresnelc(z) The Fresnel C integral obeys the mirror symmetry $\overline{C(z)} = C(\bar{z})$: >>> from sympy import conjugate >>> conjugate(fresnelc(z)) fresnelc(conjugate(z)) Differentiation with respect to $z$ is supported: >>> from sympy import diff >>> diff(fresnelc(z), z) cos(pi*z**2/2) Defining the Fresnel functions via an integral: >>> from sympy import integrate, pi, cos, expand_func >>> integrate(cos(pi*z**2/2), z) fresnelc(z)*gamma(1/4)/(4*gamma(5/4)) >>> expand_func(integrate(cos(pi*z**2/2), z)) fresnelc(z) We can numerically evaluate the Fresnel integral to arbitrary precision on the whole complex plane: >>> fresnelc(2).evalf(30) 0.488253406075340754500223503357 >>> fresnelc(-2*I).evalf(30) -0.488253406075340754500223503357*I See Also ======== fresnels: Fresnel sine integral. References ========== .. [1] https://en.wikipedia.org/wiki/Fresnel_integral .. [2] https://dlmf.nist.gov/7 .. [3] https://mathworld.wolfram.com/FresnelIntegrals.html .. [4] https://functions.wolfram.com/GammaBetaErf/FresnelC .. [5] The converging factors for the fresnel integrals by John W. 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