U -e-@sdZddlmZddlmZddlmZddlmZm Z m Z ddl m Z m Z mZddlmZddlmZdd lmZdd lmZdd lmZdd lmZmZmZdd lmZmZm Z ddl!m"Z"m#Z#ddl$m%Z%m&Z&ddl'm(Z(m)Z)m*Z*ddl+m,Z,m-Z-ddl.m/Z/m0Z0m1Z1ddl2m3Z3m4Z4dDddZ5GdddeZ6GdddeZ7GdddeZ8GdddeZ9Gdd d eZ:Gd!d"d"eZ;Gd#d$d$eZd)d*Z?Gd+d,d,eZ@Gd-d.d.eZAGd/d0d0eZBGd1d2d2eBZCGd3d4d4eBZDGd5d6d6eBZEGd7d8d8eBZFGd9d:d:eZGGd;d<dd>eGZIGd?d@d@eZJGdAdBdBeZKdCS)Ez This module contains various functions that are special cases of incomplete gamma functions. It should probably be renamed. ) EulerGamma)Add)cacheit)FunctionArgumentIndexError expand_mul)IpiRational)is_eq)Pow)S)Symbol)sympify) factorial factorial2RisingFactorial) polar_liftre unpolarify)ceilingfloor)sqrtroot)explog exp_polar)coshsinh)cossinsinc)hypermeijergTcKs|jdjr6|r,d|d<|j|f|tjfS|tjfS|rX|jdj|f|\}}n|jd\}}||t|||t|d}||t|||t|dt}||fS)NrFcomplex)argsis_extended_realexpandr Zero as_real_imagfuncr)selfdeephintsxyrZimr1h/var/www/html/Darija-Ai-Train/env/lib/python3.8/site-packages/sympy/functions/special/error_functions.pyreal_to_real_as_real_imags  (,r3cseZdZdZdZd,ddZd-ddZedd Ze e d d Z d d Z ddZ ddZddZddZddZddZddZddZddZd.d!d"Zd#d$Zd%d&Zd/d(d)Zfd*d+ZeZZS)0erfa. 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 TcCs6|dkr(dt|jdd ttSt||dSNr5r%rrr&rr rr,argindexr1r1r2fdiffs z erf.fdiffcCstSz8 Returns the inverse of this function. erfinvr8r1r1r2inversesz erf.inversecCs|jrB|tjkrtjS|tjkr&tjS|tjkr6tjS|jrBtjSt |t rV|j dSt |t rptj|j dS|jr|tjSt |t r|j djr|j dS|t}|tjtjfkr|S|r||  SdSNrr5) is_Numberr NaNInfinityOneNegativeInfinity NegativeOneis_zeror) isinstancer=r&erfcinverf2invextract_multiplicativelyrcould_extract_minus_signclsargtr1r1r2evals,        zerf.evalcGs|dks|ddkrtjSt|}t|dtd}t|dkrd|d |d|d||Sdtj||||t|ttSdSNrr%r5) r r)rrlenrErrr nr/previous_termskr1r1r2 taylor_terms "zerf.taylor_termcCs||jdSNrr+r& conjugater,r1r1r2_eval_conjugateszerf._eval_conjugatecCs |jdjSrYr&r'r\r1r1r2 _eval_is_realszerf._eval_is_realcCs |jdjrdS|jdjSdSNrT)r& is_finiter'r\r1r1r2_eval_is_finites zerf._eval_is_finitecCs |jdjSrYr&rFr\r1r1r2 _eval_is_zeroszerf._eval_is_zerocKs:ddlm}t|d|tj|tj|dttSNr uppergammar%'sympy.functions.special.gamma_functionsrgrr rCHalfr r,zkwargsrgr1r1r2_eval_rewrite_as_uppergammas zerf._eval_rewrite_as_uppergammacKs4tjt|tt}tjtt|tt|SNr rCrrr fresnelcfresnelsr,rlrmrNr1r1r2_eval_rewrite_as_fresnelsszerf._eval_rewrite_as_fresnelscKs4tjt|tt}tjtt|tt|Srorprsr1r1r2_eval_rewrite_as_fresnelcszerf._eval_rewrite_as_fresnelccKs.|ttttjggdgtddg|dSNrr%rr r#r rjr r,rlrmr1r1r2_eval_rewrite_as_meijergszerf._eval_rewrite_as_meijergcKs.d|ttttjgdtjg|d SNr%rr r"r rjryr1r1r2_eval_rewrite_as_hyperszerf._eval_rewrite_as_hypercKs,t|d||ttj|dttSNr%rexpintr rjr ryr1r1r2_eval_rewrite_as_expintszerf._eval_rewrite_as_expintNcKsbddlm}|rF|||tj}|tjkrFtjt| t|d Stjt|t|d S)Nr)limitr%) Zsympy.series.limitsrr rBrDrE_erfsrrC)r,rllimitvarrmrZlimr1r1r2_eval_rewrite_as_tractables   zerf._eval_rewrite_as_tractablecKstjt|Sro)r rCerfcryr1r1r2_eval_rewrite_as_erfcszerf._eval_rewrite_as_erfccKst tt|Srorerfiryr1r1r2_eval_rewrite_as_erfiszerf._eval_rewrite_as_erfircCsv|jdj|||d}||d}|tjkrH|j|d|dkr@dndd}||jkrh|jrhd|tt S| |SdS)Nrlogxcdirrw-+dirr%) r&as_leading_termsubsr ComplexInfinityr free_symbolsrFrr r+r,r/rrrNarg0r1r1r2_eval_as_leading_terms  zerf._eval_as_leading_termc sddlm}|d}|tjtjfkr|jdz|\}}Wnttfk r\|YSX| }|j rt ||} fddt | D|d| |g} tj t d ttt| Stt|||||S)NrOrdercs>g|]6}tj|td|dd|dd|qSr%r5)r rEr.0rWrlr1r2 sz%erf._eval_aseries..r5r%)sympy.series.orderrr rBrDr&Zleadterm ValueErrorNotImplementedError is_positiverrangerCrrr rsuperr4 _eval_aseries) r,rUargs0r/rrpoint_exZnewns __class__rr2rs$     $zerf._eval_aseries)r5)r5)N)Nr)__name__ __module__ __qualname____doc__ unbranchedr:r> classmethodrP staticmethodrrXr]r_rbrdrnrtrurzr~rrrrrrr3r* __classcell__r1r1rr2r40s2L   !   r4c@seZdZdZdZd*ddZd+ddZedd Ze e d d Z d d Z ddZ d,ddZddZddZddZddZddZddZdd Zd!d"Zd#d$Zd-d&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 Tr5cCs6|dkr(dt|jdd ttSt||dS)Nr5rRrr%r7r8r1r1r2r:`s z erfc.fdiffcCstSr;rHr8r1r1r2r>fsz erfc.inversecCs|jr2|tjkrtjS|tjkr&tjS|jr2tjSt|trLtj|j dSt|t r`|j dS|jrltjS| t }|tjtj fkr| S|rd|| SdS)Nrr%)r@r rArBr)rFrCrGr=r&rHrJrrDrKrLr1r1r2rPms$      z erfc.evalcGs|dkrtjS|dks"|ddkr(tjSt|}t|dtd}t|dkrr|d |d|d||Sdtj||||t|tt SdSrQ) r rCr)rrrSrErrr rTr1r1r2rXs "zerfc.taylor_termcCs||jdSrYrZr\r1r1r2r]szerfc._eval_conjugatecCs |jdjSrYr^r\r1r1r2r_szerfc._eval_is_realNcKs|tjdd|dSN tractableT)r-rrewriter4r,rlrrmr1r1r2rszerfc._eval_rewrite_as_tractablecKstjt|Sro)r rCr4ryr1r1r2_eval_rewrite_as_erfszerfc._eval_rewrite_as_erfcKstjttt|Sro)r rCrrryr1r1r2rszerfc._eval_rewrite_as_erficKs:tjt|tt}tjtjtt|tt|Srorprsr1r1r2rtszerfc._eval_rewrite_as_fresnelscKs:tjt|tt}tjtjtt|tt|Srorprsr1r1r2ruszerfc._eval_rewrite_as_fresnelcc Ks4tj|ttttjggdgtddg|dSrv)r rCrr r#rjr ryr1r1r2rzszerfc._eval_rewrite_as_meijergcKs4tjd|ttttjgdtjg|d Sr{)r rCrr r"rjryr1r1r2r~szerfc._eval_rewrite_as_hypercKs@ddlm}tjt|d|tj|tj|dttSre)rirgr rCrrjr rkr1r1r2rns z erfc._eval_rewrite_as_uppergammacKs2tjt|d||ttj|dttSr)r rCrrrjr ryr1r1r2rszerfc._eval_rewrite_as_expintcKs |tSrorr,r.r1r1r2_eval_expand_funcszerfc._eval_expand_funcrcCsb|jdj|||d}||d}|tjkrH|j|d|dkr@dndd}|jrTtjS||SdS)Nrrrwrrr) r&rrr rrrFrCr+rr1r1r2rs  zerfc._eval_as_leading_termcCstjt|j||||Sro)r rCr4r&r)r,rUrr/rr1r1r2rszerfc._eval_aseries)r5)r5)N)Nr)rrrrrr:r>rrPrrrXr]r_rrrrtrurzr~rnrrrr3r*rr1r1r1r2rs0L      rcseZdZdZdZd*ddZeddZee dd Z d d Z d d Z ddZ d+ddZddZddZddZddZddZddZdd Zd!d"Zd#d$ZeZd,d&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 Tr5cCs4|dkr&dt|jddttSt||dSr6r7r8r1r1r2r:sz erfi.fdiffcCs|jr2|tjkrtjS|jr"tjS|tjkr2tjS|jr>tjS|rR||  S|t}|dk r|tjkrrtSt |t rt|j dSt |t rttj |j dSt |tr|j djrt|j dSdSr?)r@r rArFr)rBrKrJrrGr=r&rHrCrIrMrlnzr1r1r2rP"s*       z erfi.evalcGs|dks|ddkrtjSt|}t|dtd}t|dkrb|d|d|d||Sd|||t|ttSdSrQ)r r)rrrSrrr rTr1r1r2rX@s  zerfi.taylor_termcCs||jdSrYrZr\r1r1r2r]Mszerfi._eval_conjugatecCs |jdjSrYr^r\r1r1r2_eval_is_extended_realPszerfi._eval_is_extended_realcCs |jdjSrYrcr\r1r1r2rdSszerfi._eval_is_zeroNcKs|tjdd|dSrrrr1r1r2rVszerfi._eval_rewrite_as_tractablecKst tt|Sro)rr4ryr1r1r2rYszerfi._eval_rewrite_as_erfcKsttt|tSro)rrryr1r1r2r\szerfi._eval_rewrite_as_erfccKs4tjt|tt}tjtt|tt|Srorprsr1r1r2rt_szerfi._eval_rewrite_as_fresnelscKs4tjt|tt}tjtt|tt|Srorprsr1r1r2rucszerfi._eval_rewrite_as_fresnelccKs0|ttttjggdgtddg|d Srvrxryr1r1r2rzgszerfi._eval_rewrite_as_meijergcKs,d|ttttjgdtjg|dSr{r}ryr1r1r2r~jszerfi._eval_rewrite_as_hypercKs>ddlm}t|d ||tj|d tttjSre)rirgrr rjr rCrkr1r1r2rnms z erfi._eval_rewrite_as_uppergammacKs0t|d ||ttj|d ttSrrryr1r1r2rqszerfi._eval_rewrite_as_expintcKs |tSrorrr1r1r2rtszerfi._eval_expand_funcrcCs\|jdj|||d}||d}||jkrB|jrBd|ttS|jrR||S||S)Nrrr%) r&rrrrFrr rar+rr1r1r2rys  zerfi._eval_as_leading_termcsddlm}|d}|tjkrt|jdfddt|D|d||g}t tdtt t |St t | ||||S)Nrrcs4g|],}td|dd|d|dqSr)rrrr1r2rsz&erfi._eval_aseries..r5r%)rrr rBr&rrrrr rrrrr,rUrr/rrrrrrr2rs    "zerfi._eval_aseries)r5)N)Nr)rrrrrr:rrPrrrXr]rrdrrrrtrurzr~rnrrr3r*rrrr1r1rr2rs0I     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/ cCs\|j\}}|dkr,dt|d ttS|dkrNdt|d ttSt||dS)Nr5rRr%)r&rrr rr,r9r/r0r1r1r2r:s  z erf2.fdiffcCstjtjtjf}|tjks$|tjkr*tjS||kr8tjS||ksH||krXt|t|St|trz|jd|krz|jdS|j s|j s|j r|j s|j r|j rt|t|S| }| }|r|r|| |  S|s|rt|t|SdSr?) r rBrDr)rAr4rGrIr&rFr' is_infiniterK)rMr/r0ZchkZsign_xZsign_yr1r1r2rPs* z erf2.evalcCs ||jd|jdSr?rZr\r1r1r2r]szerf2._eval_conjugatecCs|jdjo|jdjSr?r^r\r1r1r2rszerf2._eval_is_extended_realcKst|t|Sror4r,r/r0rmr1r1r2rszerf2._eval_rewrite_as_erfcKst|t|Srorrr1r1r2rszerf2._eval_rewrite_as_erfccKsttt|tt|Srorrr1r1r2rszerf2._eval_rewrite_as_erficKst|tt|tSro)r4rrrrr1r1r2rtszerf2._eval_rewrite_as_fresnelscKst|tt|tSro)r4rrqrr1r1r2ru szerf2._eval_rewrite_as_fresnelccKst|tt|tSro)r4rr#rr1r1r2rz szerf2._eval_rewrite_as_meijergcKst|tt|tSro)r4rr"rr1r1r2r~szerf2._eval_rewrite_as_hypercKshddlm}t|d|tj|tj|dttt|d|tj|tj|dttSrerh)r,r/r0rmrgr1r1r2rns ,,z erf2._eval_rewrite_as_uppergammacKst|tt|tSro)r4rrrr1r1r2rszerf2._eval_rewrite_as_expintcKs |tSrorrr1r1r2rszerf2._eval_expand_funccCs t|jSro)r r&r\r1r1r2rdszerf2._eval_is_zeroN)rrrrr:rrPr]rrrrrtrurzr~rnrrrdr1r1r1r2rs"E  rc@s@eZdZdZdddZdddZeddZd d Zd d Z d S)r=aR 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/ r5cCs<|dkr.ttt||jddtjSt||dSNr5rr%rr rr+r&r rjrr8r1r1r2r:Ts&z erfinv.fdiffcCstSr;rr8r1r1r2r>Zszerfinv.inversecCs|tjkrtjS|tjkr tjS|jr,tjS|tjkrrrPrrdr1r1r1r2r=!s2   r=c@sHeZdZdZdddZdddZeddZd d Zd d Z d dZ dS)rHa 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/ r5cCs>|dkr0tt t||jddtjSt||dSrrr8r1r1r2r:s(z erfcinv.fdiffcCstSr;rr8r1r1r2r>szerfcinv.inversecCsJ|tjkrtjS|jrtjS|tjkr,tjS|dkr:tjS|jrFtjSdSr)r rArFrBrCr)rDrMrlr1r1r2rPs  z erfcinv.evalcKs td|Srr<ryr1r1r2_eval_rewrite_as_erfinvszerfcinv._eval_rewrite_as_erfinvcCs|jddjSr?rcr\r1r1r2rdszerfcinv._eval_is_zerocCs |jdjSrYrcr\r1r1r2_eval_is_infiniteszerfcinv._eval_is_infiniteN)r5)r5) rrrrr:r>rrPrrdrr1r1r1r2rH~s,   rHc@s,eZdZdZddZeddZddZdS) rIa2 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/ cCsf|j\}}|dkr.t|||d|dS|dkrXtttjt|||dSt||dS)Nr5r%)r&rr+rr r rjrrr1r1r2r:s  "z erf2inv.fdiffcCs|tjks|tjkrtjS|jr,|jr,tjS|jrB|tjkrBtjS|tjkrX|jrXtjS|jrft|S|tjkrzt| S|jr|S|tjkrt|S|jr|jrtjSt|S|jr|SdSro)r rArFr)rCrBr=rH)rMr/r0r1r1r2rP s,    z erf2inv.evalcCs|j\}}|jr|jrdSdS)NTrc)r,r/r0r1r1r2rd(s  zerf2inv._eval_is_zeroN)rrrrr:rrPrdr1r1r1r2rIs 3  rIcseZdZdZeddZdddZddZd d Zd d Z d dZ ddZ e Z e Z e ZdddZdfdd Zdfdd 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|jr tjS|tjkrtjS|tjkr,tjS|jr8tjS|\}}|r`t|dtt|SdSr) rFr rDrBr)extract_branch_factorrrr rMrlrrUr1r1r2rPs   zEi.evalr5cCs0t|jd}|dkr"t||St||dSr?)rr&rrr,r9rNr1r1r2r:s zEi.fdiffcCs:|jdtdjr.t||tt|St||Sr)r&rrr _eval_evalfrr r,precr1r1r2rszEi._eval_evalfcKs(ddlm}|dtd| ttS)Nrrfrw)rirgrrr rkr1r1r2rns zEi._eval_rewrite_as_uppergammacKstdtd| ttS)Nr5rw)rrrr ryr1r1r2rszEi._eval_rewrite_as_expintcKs$t|trt|jdStt|SrY)rGrlir&rryr1r1r2_eval_rewrite_as_lis zEi._eval_rewrite_as_licKs2|jrt|t|ttSt|t|SdSro) is_negativeShiChirr ryr1r1r2_eval_rewrite_as_SiszEi._eval_rewrite_as_SiNcKst|t|Sro)r_eisrr1r1r2rszEi._eval_rewrite_as_tractablerc sddlm}|jd|d}|jdj||d}|||}|jr||\}}|dkrbt|n|}t|||t ||j rt t nt jStj|||dS)Nr)r)rr)Zsympyrr&rrrrFas_coeff_exponentrrrrr r r)rr) r,r/rrrx0rNcerr1r2rs  zEi._eval_as_leading_termcsB|jd|d}|jr2|j|j}||||St|||SrY)r&rrFr _eval_nseriesrr,r/rUrrrfrr1r2rs  zEi._eval_nseriescs|ddlm}|d}|tjkrf|jdfddt|D|d||g}tt|Stt | ||||S)Nrrcsg|]}t||qSr1rrrr1r2rsz$Ei._eval_aseries..r5) rrr rBr&rrrrrrrrrr2rs   zEi._eval_aseries)r5)N)Nr)r)rrrrrrPr:rrnrrr_eval_rewrite_as_Ci_eval_rewrite_as_Chi_eval_rewrite_as_Shirrrrrr1r1rr2r1s W     rcsneZdZdZeddZddZddZdd Zd d Z d d Z e Z e Z e Z dfdd ZfddZZS)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|}||kr*t||S|jr8|dksH|jsjd|jrjtt||d|d||S|\}}|tj krdS|j r|dksdSt||dt t |tj |dt|dt||dStdt t ||d||d|d|t||SdS)Nr)gammargr%r5)rirrgrr is_Integerrrr r) is_integerr rrErr)rMnurlrrgZnu2rUr1r1r2rPKs  "  8z expint.evalcCsd|j\}}|dkr>||d tgddgddd|gg|S|dkrVt|d| St||dSr)r&r#rr)r,r9rrlr1r1r2r:as  ,z expint.fdiffcKs&ddlm}||d|d||S)Nrrfr5)rirg)r,rrlrmrgr1r1r2rnjs z"expint._eval_rewrite_as_uppergammac sdkr(t|tt t ttSjrdkrt| dtdt|tt tdt fddt dDS|SdS)Nr5cs$g|]}t|d|qSr%rrrr/r1r2rvsz.expint._eval_rewrite_as_Ei..) rrrr rrrE1rrrrr,rrlrmr1rr2_eval_rewrite_as_Eins  $zexpint._eval_rewrite_as_EicKs|tjtf|Sro)rrrrr1r1r2rzszexpint._eval_expand_funccKs|dkr |St|t|Sr)rrrr1r1r2r}szexpint._eval_rewrite_as_Sircst|jd|sd|jd}|dkr<|j|j}||||S|jrd|dkrd|j|j}||||St|||Sr?)r&hasrrrrr)r,r/rUrrrrrr1r2rs   zexpint._eval_nseriescsddlm}|d}|jd|tjkrt|jdfddt|D|d||g}t t|Stt | ||||S)Nrrr5cs(g|] }tj|t||qSr1)r rErrrrlr1r2rsz(expint._eval_aseries..) rrr&r rBrrrrrrrrrr2rs    ,zexpint._eval_aseries)r)rrrrrrPr:rnrrrrrrrrrr1r1rr2rsg    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. r5)rrr1r1r2rs rc@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<|jr tjS|tjkrtjS|tjkr,tjS|jr8tjSdSro)rFr r)rCrDrBrr1r1r2rP$s  zli.evalr5cCs.|jd}|dkr tjt|St||dSr?r&r rCrrrr1r1r2r:/s zli.fdiffcCs"|jd}|js||SdSrY)r&Zis_extended_negativer+r[r,rlr1r1r2r]6s zli._eval_conjugatecKst|tdSr)Lirryr1r1r2_eval_rewrite_as_Li<szli._eval_rewrite_as_LicKs tt|Sro)rrryr1r1r2r?szli._eval_rewrite_as_EicKsPddlm}|dt|  tjtt|ttjt|tt| SNrrf)rirgrr rjrCrkr1r1r2rnBs  " zli._eval_rewrite_as_uppergammacKsXttt|tttt|tjttjt|tt|ttt|Sro)CirrSir rjrCryr1r1r2rGs ""zli._eval_rewrite_as_SicKs<tt|tt|tjttjt|tt|Sro)rrrr rjrCryr1r1r2rMszli._eval_rewrite_as_ShicKs@t|tddt|tjtt|ttjt|tS)N)r5r5)r%r%)rr"r rjrCrryr1r1r2r~Rs "zli._eval_rewrite_as_hypercKsFtt|  tjttjt|tt|tddt| S)N)r1r5))rrr1)rr rjrCr#ryr1r1r2rzVs2zli._eval_rewrite_as_meijergNcKs|tt|Sro)rrrr1r1r2rZszli._eval_rewrite_as_tractablercs:|jdfddtd|D}tttt|S)Nrcs$g|]}t|t||qSr1)rrrrr1r2r_sz$li._eval_nseries..r5)r&rrrr)r,r/rUrrrr1rr2r]s zli._eval_nseriescCs|jd}|jrdSdSr`rcrr1r1r2rdbs zli._eval_is_zero)r5)N)r)rrrrrrPr:r]rrrnrrrrr~rzrrrdr1r1r1r2rs"c   rc@sJeZdZdZeddZdddZddZd d Zdd d Z dddZ d S)rab 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&|tjkrtjS|tdkr"tjSdSr)r rBr)rr1r1r2rPs  zLi.evalr5cCs.|jd}|dkr tjt|St||dSr?rrr1r1r2r:s zLi.fdiffcCs|t|Sro)rrZevalfrr1r1r2rszLi._eval_evalfcKst|tdSr)rryr1r1r2rszLi._eval_rewrite_as_liNcKs|tjdddS)NrT)r-)rrrr1r1r2rszLi._eval_rewrite_as_tractablercCs|j|j}||||Sro)rr&r)r,r/rUrrrr1r1r2rs zLi._eval_nseries)r5)N)r) rrrrrrPr:rrrrr1r1r1r2rgs@   rcsHeZdZdZeddZdddZddZd d Zdfd d Z Z S)TrigonometricIntegralz) Base class for trigonometric integrals. cCs,|tjkr|jS|tjkr"|S|tjkr4|S|jr@|jS|t t }|dkrn| ddkrn|t }|dk r| |dS|t t }|dk r| |dS|t d}|dkr| ddkr|d}|dk r| |S|\}}|dkr ||kr dSdtt || d||S)Nrr5rwr%)r r)_atzerorB_atinfrD _atneginfrFrJrr _trigfunc_Ifactor _minusfactorrr rr1r1r2rPs2         zTrigonometricIntegral.evalr5cCs2t|jd}|dkr$|||St||dSr?)rr&rrrr1r1r2r:szTrigonometricIntegral.fdiffcKs||tSro)rrrryr1r1r2rsz)TrigonometricIntegral._eval_rewrite_as_EicKsddlm}|||Sr)rirgrrrkr1r1r2rns z1TrigonometricIntegral._eval_rewrite_as_uppergammarcs|d7}|jd|ddkr.t|||S|||||}|ddkrX|d8}|jtdddd}|ddkr|tt|7}|||jd|||S)Nr5rcSs |||Sror1)rOrUr1r1r2z5TrigonometricIntegral._eval_nseries..F)Z simultaneous) r&rrrrreplacer rr)r,r/rUrrZ baseseriesrr1r2rsz#TrigonometricIntegral._eval_nseries)r5)r) rrrrrrPr:rrnrrr1r1rr2rs  rcsreZdZdZeZejZe ddZ e ddZ e ddZ e dd Z d d Zd d ZfddZddZZS)ra 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 ttjSror r rjrMr1r1r2rQsz Si._atinfcCs t tjSrorr r1r1r2rUsz Si._atneginfcCs t| Sro)rrr1r1r2rYszSi._minusfactorcCstt||Sro)rrrMrlsignr1r1r2r]sz Si._IfactorcKs2tdttt|ttt |dtSr)r rrrryr1r1r2raszSi._eval_rewrite_as_expintcKs,ddlm}tddd}|t||d|fS)Nr)IntegralrOT)ZDummy)Zsympy.integrals.integralsr rr!)r,rlrmr rOr1r1r2_eval_rewrite_as_sinces  zSi._eval_rewrite_as_sincc sddlm}|d}|tjkr|jdfddtt|ddD|d||g}fddtt|ddD|d||g}tdtt |t t |St t | ||||S)Nrrcs.g|]&}tj|td|d|qSrr rErrrr1r2rqsz$Si._eval_aseries..r5r%cs6g|].}tj|td|dd|dqSrrrrr1r2rss)rrr rBr&rintr rrr rrr r,rUrr/rrrpqrrr2rjs      0zSi._eval_aseriescCs|jd}|jrdSdSr`rcrr1r1r2rdzs zSi._eval_is_zero)rrrrr rr r)rrrrrrrr rrdrr1r1rr2rsF     rcsleZdZdZeZejZe ddZ e ddZ e ddZ e dd Z d d ZdddZfddZZS)ra 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 cCstjSro)r r)r r1r1r2rsz Ci._atinfcCsttSro)rr r r1r1r2rsz Ci._atneginfcCst|ttSrorrr rr1r1r2rszCi._minusfactorcCst|ttd|Srrrr r r1r1r2rsz Ci._IfactorcKs(ttt|ttt | dSr)rrrryr1r1r2rszCi._eval_rewrite_as_expintNrcCs|jdj|||d}||d}|tjkrJ|j|dt|jrBdndd}|jr| |\}}|dkrnt |n|}t |||t S|j r| |S|SdSNrrrrrr&rrr rArrrrFrrrrar+r,r/rrrNrrrr1r1r2rs   zCi._eval_as_leading_termc sddlm}|d}|tjkr|jdfddtt|ddD|d||g}fddtt|ddD|d||g}tt|t t|St t | ||||S)Nrrcs.g|]&}tj|td|d|qSrrrrr1r2rsz$Ci._eval_aseries..r5r%cs6g|].}tj|td|dd|dqSrrrrr1r2rs) rrr rBr&rrr rrrrrrrrr2rs      (zCi._eval_aseries)Nr)rrrrrrr rrrrrrrrrrrr1r1rr2rsO     rc@sdeZdZdZeZejZe ddZ e ddZ e ddZ e dd Z d d Zd d ZdddZdS)ra 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 cCstjSror rBr r1r1r2rIsz Shi._atinfcCstjSro)r rDr r1r1r2rMsz Shi._atneginfcCs t| Sro)rrr1r1r2rQszShi._minusfactorcCstt||Sro)rrr r1r1r2rUsz Shi._IfactorcKs,t|tttt|dttdSr)rrrr ryr1r1r2rYszShi._eval_rewrite_as_expintcCs|jd}|jrdSdSr`rcrr1r1r2rd]s zShi._eval_is_zeroNrcCsf|jd|}||d}|tjkrD|j|dt|jr>> 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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