o *ήc -@sZdZddlmZmZmZmZmZmZmZm Z ddl m Z m Z m Z ddlmZddlmZddlmZddlmZmZmZddlmZmZmZdd lmZmZdd lm Z m!Z!dd l"m#Z#m$Z$m%Z%dd l&m'Z'm(Z(dd l)m*Z*m+Z+m,Z,ddl-m.Z.m/Z/d?ddZ0Gddde Z1Gddde Z2Gddde Z3Gddde Z4Gddde Z5Gddde Z6Gddde Z7Gd d!d!e Z8Gd"d#d#e Z9d$d%Z:Gd&d'd'e Z;Gd(d)d)e ZGd.d/d/e=Z?Gd0d1d1e=Z@Gd2d3d3e=ZAGd4d5d5e ZBGd6d7d7eBZCGd8d9d9eBZDGd:d;d;e ZEGdS)@z This module contains various functions that are special cases of incomplete gamma functions. It should probably be renamed. )AddSsympifycacheitpiIRational EulerGamma)FunctionArgumentIndexError expand_mul)is_eq)Pow)Symbol) factorial factorial2RisingFactorial) polar_liftre unpolarify)ceilingfloor)sqrtroot)explog exp_polar)coshsinh)cossinsinc)hypermeijergTcKs|jdjr|rd|d<|j|fi|tjfS|tjfS|r0|jdj|fi|\}}n |jd\}}||t|||t|d}||t|||t|dt}||fS)NrFcomplex)argsis_extended_realexpandrZero as_real_imagfuncr)selfdeephintsxyrimr2N/tmp/pip-target-vg8gfxp4/lib/python/sympy/functions/special/error_functions.pyreal_to_real_as_real_imags  "(,r4cseZdZdZdZd,ddZd,ddZedd Ze e d d Z d d Z ddZ ddZddZddZddZddZddZddZddZd-d!d"Zd#d$Zd%d&Zd.d(d)Zfd*d+ZeZZS)/erfa+ 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] http://dlmf.nist.gov/7 .. [3] http://mathworld.wolfram.com/Erf.html .. [4] http://functions.wolfram.com/GammaBetaErf/Erf TcCs4|dkrdt|jdd ttjSt||Nr6r%rrr&rrPir r,argindexr2r2r3fdiffz" z erf.fdiffcCtSz8 Returns the inverse of this function. erfinvr:r2r2r3inversez 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|tj}|tjtjfvr^|S|rh||  SdSNrr6) is_NumberrNaNInfinityOneNegativeInfinity NegativeOneis_zeror) isinstancerAr&erfcinverf2invextract_multiplicatively ImaginaryUnitcould_extract_minus_signclsargtr2r2r3evals.         zerf.evalcGs|dks |ddkr tjSt|}t|dtd}t|dkr2|d |d|d||Sdtj||||t|ttjSNrr%r6) rr)rrlenrJrrr9nr/previous_termskr2r2r3 taylor_terms ",zerf.taylor_termcC||jdSNrr+r& conjugater,r2r2r3_eval_conjugatezerf._eval_conjugatecC |jdjSr`r&r'rcr2r2r3 _eval_is_real zerf._eval_is_realcCs|jdjrdS|jdjSNrT)r& is_finiter'rcr2r2r3_eval_is_finites  zerf._eval_is_finitecCrfr`r&rKrcr2r2r3 _eval_is_zerorizerf._eval_is_zerocKs<ddlm}t|d|tj|tj|dttjSNr uppergammar%'sympy.functions.special.gamma_functionsrqrrrHHalfr9r,zkwargsrqr2r2r3_eval_rewrite_as_uppergammas 0zerf._eval_rewrite_as_uppergammacK8tjtj|tt}tjtjt|tt|SNrrHrPrrfresnelcrfresnelsr,rvrwrTr2r2r3_eval_rewrite_as_fresnels zerf._eval_rewrite_as_fresnelscKryrzr{r~r2r2r3_eval_rewrite_as_fresnelcrzerf._eval_rewrite_as_fresnelccKs.|ttttjggdgtddg|dSNrr%rrr#rrtrr,rvrwr2r2r3_eval_rewrite_as_meijerg.zerf._eval_rewrite_as_meijergcKs.d|ttttjgdtjg|d SNr%rrr"rrtrr2r2r3_eval_rewrite_as_hyperrzerf._eval_rewrite_as_hypercKs.t|d||ttj|dttjSNr%rexpintrrtr9rr2r2r3_eval_rewrite_as_expintrzerf._eval_rewrite_as_expintNcKsbddlm}|r#|||tj}|tjur#tjt| t|d Stjt|t|d S)Nr)limitr%) sympy.series.limitsrrrGrIrJ_erfsrrH)r,rvlimitvarrwrlimr2r2r3_eval_rewrite_as_tractables  zerf._eval_rewrite_as_tractablecKtjt|Srz)rrHerfcrr2r2r3_eval_rewrite_as_erfczerf._eval_rewrite_as_erfccKt tt|Srzrerfirr2r2r3_eval_rewrite_as_erfizerf._eval_rewrite_as_erfircCsr|jdj|||d}||d}|tjur$|j|d|dkr dndd}||jvr4|jr4d|tt S| |S)Nrlogxcdirr-+dirr%) r&as_leading_termsubsrComplexInfinityr free_symbolsrKrrr+r,r/rrrTarg0r2r2r3_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%r6)rrJr.0r]rvr2r3 s6z%erf._eval_aseries..r6r%)sympy.series.orderrrrGrIr&leadterm ValueErrorNotImplementedError is_positiverrangerHrrrrsuperr5 _eval_aseries) r,r[args0r/rrpoint_exnewns __class__rr3rs&    $zerf._eval_aseriesr6rzr`)__name__ __module__ __qualname____doc__ unbranchedr<rB classmethodrV staticmethodrr^rdrhrlrnrxrrrrrrrrrrr4r* __classcell__r2r2rr3r5+s4L   !     r5c@seZdZdZdZd*ddZd*ddZedd Ze e d d Z d d Z ddZ d+ddZddZddZddZddZddZddZdd Zd!d"Zd#d$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] http://dlmf.nist.gov/7 .. [3] http://mathworld.wolfram.com/Erfc.html .. [4] http://functions.wolfram.com/GammaBetaErf/Erfc Tr6cCs4|dkrdt|jdd ttjSt||)Nr6rXrr%r8r:r2r2r3r<[r=z erfc.fdiffcCr>r?rMr:r2r2r3rBarCz erfc.inversecCs|jr|tjur tjS|tjurtjS|jrtjSt|tr&tj|j dSt|t r0|j dS|jr6tjS| tj }|tjtj fvrG| S|rRd|| SdS)Nrr%)rErrFrGr)rKrHrLrAr&rMrOrPrIrQrRr2r2r3rVhs&      z erfc.evalcGs|dkrtjS|dks|ddkrtjSt|}t|dtd}t|dkr9|d |d|d||Sdtj||||t|ttj SrW) rrHr)rrrYrJrrr9rZr2r2r3r^s ",zerfc.taylor_termcCr_r`rarcr2r2r3rdrezerfc._eval_conjugatecCrfr`rgrcr2r2r3rhrizerfc._eval_is_realNcK|tjdd|dSN tractableT)r-rrewriter5r,rvrrwr2r2r3rzerfc._eval_rewrite_as_tractablecKrrz)rrHr5rr2r2r3_eval_rewrite_as_erfrzerfc._eval_rewrite_as_erfcKstjttt|Srz)rrHrrrr2r2r3rrzerfc._eval_rewrite_as_erficK>tjtj|tt}tjtjtjt|tt|Srzr{r~r2r2r3r&zerfc._eval_rewrite_as_fresnelscKrrzr{r~r2r2r3rrzerfc._eval_rewrite_as_fresnelcc Ks4tj|ttttjggdgtddg|dSr)rrHrrr#rtrrr2r2r3r4zerfc._eval_rewrite_as_meijergcKs4tjd|ttttjgdtjg|d Sr)rrHrrr"rtrr2r2r3rrzerfc._eval_rewrite_as_hypercKsBddlm}tjt|d|tj|tj|dttjSro)rsrqrrHrrtr9rur2r2r3rxs 6z erfc._eval_rewrite_as_uppergammacKs4tjt|d||ttj|dttjSr)rrHrrrtr9rr2r2r3rrzerfc._eval_rewrite_as_expintcK |tSrzrr,r.r2r2r3_eval_expand_func zerfc._eval_expand_funcrcCs^|jdj|||d}||d}|tjur$|j|d|dkr dndd}|jr*tjS||S)Nrrrrrr) r&rrrrrrKrHr+rr2r2r3rs   zerfc._eval_as_leading_termcCstjt|j||||Srz)rrHr5r&r)r,r[rr/rr2r2r3rzerfc._eval_aseriesrrzr`)rrrrrr<rBrrVrrr^rdrhrrrrrrrrxrrrr4r*rr2r2r2r3r s2L        rcseZdZdZdZd*ddZeddZee dd Z d d Z d d Z ddZ d+ddZddZddZddZddZddZddZdd Zd!d"Zd#d$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] http://mathworld.wolfram.com/Erfi.html .. [3] http://functions.wolfram.com/GammaBetaErf/Erfi Tr6cCs2|dkrdt|jddttjSt||r7r8r:r2r2r3r<s  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 dSdSdSdSrD)rErrFrKr)rGrQrOrrLrAr&rMrHrNrSrvnzr2r2r3rVs.       z erfi.evalcGs|dks |ddkr tjSt|}t|dtd}t|dkr1|d|d|d||Sd|||t|ttjSrW)rr)rrrYrrr9rZr2r2r3r^;s  "zerfi.taylor_termcCr_r`rarcr2r2r3rdHrezerfi._eval_conjugatecCrfr`rgrcr2r2r3_eval_is_extended_realKrizerfi._eval_is_extended_realcCrfr`rmrcr2r2r3rnNrizerfi._eval_is_zeroNcKrrrrr2r2r3rQrzerfi._eval_rewrite_as_tractablecKrrz)rr5rr2r2r3rTrzerfi._eval_rewrite_as_erfcKsttt|tSrz)rrrr2r2r3rWrezerfi._eval_rewrite_as_erfccK8tjtj|tt}tjtjt|tt|Srzr{r~r2r2r3rZrzerfi._eval_rewrite_as_fresnelscKrrzr{r~r2r2r3r^rzerfi._eval_rewrite_as_fresnelccKs0|ttttjggdgtddg|d Srrrr2r2r3rbs0zerfi._eval_rewrite_as_meijergcKs,d|ttttjgdtjg|dSrrrr2r2r3res,zerfi._eval_rewrite_as_hypercKs@ddlm}t|d ||tj|d ttjtjSro)rsrqrrrtr9rHrur2r2r3rxhs 4z erfi._eval_rewrite_as_uppergammacKs2t|d ||ttj|d ttjSrrrr2r2r3rls2zerfi._eval_rewrite_as_expintcKrrzrrr2r2r3rorzerfi._eval_expand_funcrcCs\|jdj|||d}||d}||jvr!|jr!d|ttS|jr)||S||S)Nrrr%) r&rrrrKrrrkr+rr2r2r3rts   zerfi._eval_as_leading_termcsddlm}|d}|tjur;|jdfddt|D|d||g}tj tdtt t |St t | ||||S)Nrrcs4g|]}td|dd|d|dqSr)rrrr2r3rs,z&erfi._eval_aseries..r6r%)rrrrGr&rrPrrrrrrrr,r[rr/rrrrrrr3r~s    $zerfi._eval_aseriesrrzr`)rrrrrr<rrVrrr^rdrrnrrrrrrrrxrrr4r*rrrr2r2rr3rs2I       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] http://functions.wolfram.com/GammaBetaErf/Erf2/ cCs\|j\}}|dkrdt|d ttjS|dkr)dt|d ttjSt||)Nr6rXr%)r&rrrr9r r,r;r/r0r2r2r3r<s  z erf2.fdiffcCstjtjtjf}|tjus|tjurtjS||krtjS||vs$||vr,t|t|St|tr=|jd|kr=|jdS|j sO|j sO|j rI|j sO|j rW|j rWt|t|S| }| }|rk|rk|| |  S|so|rwt|t|SdSrD) rrGrIr)rFr5rLrNr&rKr' is_infiniterQ)rSr/r0chksign_xsign_yr2r2r3rVs, z erf2.evalcCs ||jd|jdSrDrarcr2r2r3rds zerf2._eval_conjugatecCs|jdjo |jdjSrDrgrcr2r2r3rzerf2._eval_is_extended_realcKst|t|Srzr5r,r/r0rwr2r2r3rzerf2._eval_rewrite_as_erfcKst|t|Srzrrr2r2r3rrzerf2._eval_rewrite_as_erfccKsttt|tt|Srzrrr2r2r3rrzerf2._eval_rewrite_as_erficKt|tt|tSrz)r5rr}rr2r2r3rrzerf2._eval_rewrite_as_fresnelscKrrz)r5rr|rr2r2r3rrzerf2._eval_rewrite_as_fresnelccKrrz)r5rr#rr2r2r3rrzerf2._eval_rewrite_as_meijergcKrrz)r5rr"rr2r2r3r rzerf2._eval_rewrite_as_hypercKslddlm}t|d|tj|tj|dttjt|d|tj|tj|dttjSrorr)r,r/r0rwrqr2r2r3rxs ..z erf2._eval_rewrite_as_uppergammacKrrz)r5rrrr2r2r3rrzerf2._eval_rewrite_as_expintcKrrzrrr2r2r3rrzerf2._eval_expand_funccCs t|jSrz)r r&rcr2r2r3rnrzerf2._eval_is_zeroN)rrrrr<rrVrdrrrrrrrrrxrrrnr2r2r2r3rs$E  rc@s@eZdZdZdddZdddZeddZd d Zd d Z d S)rAaQ 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] http://functions.wolfram.com/GammaBetaErf/InverseErf/ r6cCs:|dkrttjt||jddtjSt||Nr6rr%rrr9rr+r&rtr r:r2r2r3r<Os( z erfinv.fdiffcCr>r?rr:r2r2r3rBUrCzerfinv.inversecCs|tjurtjS|tjurtjS|jrtjS|tjurtjSt|t r.|j dj r.|j dS|jr4tjS| d}|durNt|t rP|j dj rR|j d SdSdSdSNrr) rrFrJrIrKr)rHrGrLr5r&r'rOrr2r2r3rV\s       z erfinv.evalcK td|SNr6rrr2r2r3_eval_rewrite_as_erfcinvrrizerfinv._eval_rewrite_as_erfcinvcCrfr`rmrcr2r2r3rnurizerfinv._eval_is_zeroNr) rrrrr<rBrrVrrnr2r2r2r3rAs 2   rAc@sHeZdZdZdddZdddZeddZd d Zd d Z d dZ dS)rMa 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] http://functions.wolfram.com/GammaBetaErf/InverseErfc/ r6cCs<|dkrttj t||jddtjSt||rrr:r2r2r3r<s* z erfcinv.fdiffcCr>r?rr:r2r2r3rBrCzerfcinv.inversecCsJ|tjurtjS|jrtjS|tjurtjS|dkrtjS|jr#tjSdSr)rrFrKrGrHr)rIrSrvr2r2r3rVs  z erfcinv.evalcKrrr@rr2r2r3_eval_rewrite_as_erfinvrizerfcinv._eval_rewrite_as_erfinvcCs|jddjSrDrmrcr2r2r3rnrzerfcinv._eval_is_zerocCrfr`rmrcr2r2r3_eval_is_infiniterizerfcinv._eval_is_infiniteNr) rrrrr<rBrrVrrnrr2r2r2r3rMys ,    rMc@s,eZdZdZddZeddZddZdS) rNa1 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] http://functions.wolfram.com/GammaBetaErf/InverseErf2/ cCsd|j\}}|dkrt|||d|dS|dkr-ttjtjt|||dSt||)Nr6r%)r&rr+rrr9rtr rr2r2r3r<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]|SdSrz)rrFrKr)rHrGrArM)rSr/r0r2r2r3rVs.    z erf2inv.evalcCs"|j\}}|jr |jrdSdSdS)NTrm)r,r/r0r2r2r3rn#s  zerf2inv._eval_is_zeroN)rrrrr<rrVrnr2r2r2r3rNs 3  rNcseZdZdZeddZdddZddZd 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] http://dlmf.nist.gov/6.6 .. [2] https://en.wikipedia.org/wiki/Exponential_integral .. [3] Abramowitz & Stegun, section 5: http://people.math.sfu.ca/~cbm/aands/page_228.htm cCsd|jrtjS|tjurtjS|tjurtjS|jrtjS|\}}|r0t|dtt|SdSr) rKrrIrGr)extract_branch_factorrrrrSrvrr[r2r2r3rVs   zEi.evalr6cCs,t|jd}|dkrt||St||rD)rr&rr r,r;rTr2r2r3r<s  zEi.fdiffcCs:|jdtdjrt||tt|St||Sr)r&rrr _eval_evalfrrr,precr2r2r3rs zEi._eval_evalfcKs(ddlm}|dtd| ttS)Nrrpr)rsrqrrrrur2r2r3rxs zEi._eval_rewrite_as_uppergammacKstdtd| ttS)Nr6r)rrrrrr2r2r3rrzEi._eval_rewrite_as_expintcKs$t|tr t|jdStt|Sr`)rLrlir&rrr2r2r3_eval_rewrite_as_lis  zEi._eval_rewrite_as_licKs.|jrt|t|ttSt|t|Srz) is_negativeShiChirrrr2r2r3_eval_rewrite_as_SiszEi._eval_rewrite_as_SiNcKst|t|Srz)r_eisrr2r2r3rrzEi._eval_rewrite_as_tractablerc sddlm}|jd|d}|jdj||d}|||}|jrL||\}}|dur1t|n|}t|||t j ||j rHt j t St jStj|||dS)Nr)r)rr)sympyrr&rrrrKas_coeff_exponentrrr rrPrr)rr) r,r/rrrx0rTcerr2r3rs  zEi._eval_as_leading_termcB|jd|d}|jr|j|j}||||St|||Sr`)r&rrKr _eval_nseriesrr,r/r[rrrfrr2r3r   zEi._eval_nseriescs|ddlm}|d}|tjur3|jdfddt|D|d||g}tt|Stt | ||||S)Nrrcsg|] }t||qSr2rrrr2r3rz$Ei._eval_aseries..r6) rrrrGr&rrrrrrrrrr3rs   zEi._eval_aseriesrrzr`r)rrrrrrVr<rrxrrr_eval_rewrite_as_Ci_eval_rewrite_as_Chi_eval_rewrite_as_Shirrr rrr2r2rr3r,s"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] http://dlmf.nist.gov/8.19 .. [2] http://functions.wolfram.com/GammaBetaErf/ExpIntegralE/ .. [3] https://en.wikipedia.org/wiki/Exponential_integral cCsddlm}m}t|}||krt||S|jr|dks$|js5d|jr5tt||d|d||S|\}}|tj urBdS|j rm|dksKdSt||dt t |tj |dt|dt||dStdt t ||d||d|d|t||S)Nr)gammarqr%r6)rsrrqrr is_Integerr rrr) is_integerrrrJrr)rSnurvrrqnu2r[r2r2r3rVFs  "  8>z expint.evalcCs`|j\}}|dkr||d tgddgddd|gg|S|dkr+t|d| St||r)r&r#rr )r,r;rrvr2r2r3r<\s , z expint.fdiffcKs&ddlm}||d|d||S)Nrrpr6)rsrq)r,rrvrwrqr2r2r3rxes z"expint._eval_rewrite_as_uppergammac sdkrt|tt t ttSjrMdkrMt| dtdt|tt tdt fddt dDS|S)Nr6cs$g|]}t|d|qSr%rrrr/r2r3rq$z.expint._eval_rewrite_as_Ei..) rrrrrrrE1rrrrr,rrvrwr2rr3_eval_rewrite_as_Eiis  $zexpint._eval_rewrite_as_EicKs|tjtfi|Srz)rrrrr2r2r3rurzexpint._eval_expand_funccKs|dkr|St|t|Sr)rrrr2r2r3rxszexpint._eval_rewrite_as_Sircst|jd|s2|jd}|dkr|j|j}||||S|jr2|dkr2|j|j}||||St|||SrD)r&hasrr rr r)r,r/r[rrrrrr2r3r s   zexpint._eval_nseriescsddlm}|d}|jd|tjur:|jdfddt|D|d||g}t t|Stt | ||||S)Nrrr6cs(g|]}tj|t||qSr2)rrJrrrrvr2r3rs(z(expint._eval_aseries..) rrr&rrGrrrrrrrrr"r3rs    ,zexpint._eval_aseriesr)rrrrrrVr<rxr rrrrrr rrr2r2rr3rsg    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. r6)rrr2r2r3rs 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] http://mathworld.wolfram.com/LogarithmicIntegral.html .. [3] http://dlmf.nist.gov/6 .. [4] http://mathworld.wolfram.com/SoldnersConstant.html cCs<|jrtjS|tjurtjS|tjurtjS|jrtjSdSrz)rKrr)rHrIrGrr2r2r3rVs  zli.evalr6cC*|jd}|dkrtjt|St||rDr&rrHrr rr2r2r3r<*  zli.fdiffcCs"|jd}|js||SdSr`)r&is_extended_negativer+rbr,rvr2r2r3rd1s zli._eval_conjugatecKst|tdSr)Lirrr2r2r3_eval_rewrite_as_Li7rzli._eval_rewrite_as_LicKs tt|Srz)rrrr2r2r3r :rizli._eval_rewrite_as_EicKsPddlm}|dt|  tjtt|ttjt|tt| SNrrp)rsrqrrrtrHrur2r2r3rx=s " zli._eval_rewrite_as_uppergammacKsXttt|tttt|tjttjt|tt|ttt|Srz)CirrSirrtrHrr2r2r3rBs ""zli._eval_rewrite_as_SicKs<tt|tt|tjttjt|tt|Srz)rrrrrtrHrr2r2r3rHs<zli._eval_rewrite_as_ShicKsBt|tddt|tjtt|ttjt|tjS)N)r6r6)r%r%)rr"rrtrHr rr2r2r3rMs "zli._eval_rewrite_as_hypercKsFtt|  tjttjt|tt|tddt| S)N)r2r))rrr2)rrrtrHr#rr2r2r3rQs2zli._eval_rewrite_as_meijergNcKs|tt|Srz)rrrr2r2r3rUrzli._eval_rewrite_as_tractablercs<|jdfddtd|D}tjttt|S)Nrcs$g|]}t|t||qSr2)rrrrr2r3rZrz$li._eval_nseries..r6)r&rrr rr)r,r/r[rrrr2rr3r Xs zli._eval_nseriescC|jd}|jr dSdSrjrmr'r2r2r3rn] zli._eval_is_zerorrzr)rrrrrrVr<rdr)r rxrrrrrrrr rnr2r2r2r3rs$c     rc@sJeZdZdZeddZdddZddZd d Zdd d Z dddZ d S)r(a` 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] http://mathworld.wolfram.com/LogarithmicIntegral.html .. [3] http://dlmf.nist.gov/6 cCs&|tjurtjS|tdkrtjSdSr)rrGr)rr2r2r3rVs  zLi.evalr6cCr#rDr$rr2r2r3r<r%zLi.fdiffcCs|t|Srz)rrevalfrr2r2r3rrzLi._eval_evalfcKst|tdSr)rrr2r2r3rrzLi._eval_rewrite_as_liNcKs|tjdddS)NrT)r-)rrrr2r2r3rrezLi._eval_rewrite_as_tractablercCs|j|j}||||Srz)rr&r )r,r/r[rrrr2r2r3r s zLi._eval_nseriesrrzr) rrrrrrVr<rrrr r2r2r2r3r(bs@   r(csHeZdZdZeddZdddZddZd 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)Nrr6rr%)rr)_atzerorG_atinfrI _atneginfrKrOrr _trigfunc_Ifactor _minusfactorrrrr2r2r3rVs2         "zTrigonometricIntegral.evalr6cCs.t|jd}|dkr|||St||rD)rr&r4r rr2r2r3r<s zTrigonometricIntegral.fdiffcKs||tSrz)rrrrr2r2r3r rz)TrigonometricIntegral._eval_rewrite_as_EicKsddlm}|||Sr*)rsrqrrrur2r2r3rxs z1TrigonometricIntegral._eval_rewrite_as_uppergammarcs|d7}|jd|ddkrt|||S|||||}|ddkr,|d8}|jtdddd}|ddkrE|tt|7}|||jd|||S)Nr6rcSs |||Srzr2)rUr[r2r2r3s z5TrigonometricIntegral._eval_nseries..F) simultaneous) r&rrr r4replacerr r)r,r/r[rr baseseriesrr2r3r sz#TrigonometricIntegral._eval_nseriesrr) rrrrrrVr<r rxr rr2r2rr3r0s  r0csreZdZdZeZejZe ddZ e ddZ e ddZ e dd Z d d Zd d ZfddZddZZS)r,a 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 ttjSrzrrrtrSr2r2r3r2L z Si._atinfcCs t tjSrzr;r<r2r2r3r3Ps z Si._atneginfcC t| Srz)r,rr2r2r3r6Tr=zSi._minusfactorcCtt||Srz)rrrSrvsignr2r2r3r5Xz Si._IfactorcKs2tdttt|ttt |dtSr)rrrrrr2r2r3r\s2zSi._eval_rewrite_as_expintcKs,ddlm}tddd}|t||d|fS)Nr)IntegralrUT)Dummy)sympy.integrals.integralsrCrr!)r,rvrwrCrUr2r2r3_eval_rewrite_as_sinc`s  zSi._eval_rewrite_as_sincc sddlm}|d}|tjurb|jdfddtt|ddD|d||g}fddtt|ddD|d||g}tdtt |t t |St t | ||||S)Nrrc.g|]}tj|td|d|qSrrrJrrrr2r3rl&z$Si._eval_aseries..r6r%c6g|]}tj|td|dd|dqSrrHrrr2r3rn.)rrrrGr&rintrrrr rr,r r,r[rr/rrrpqrrr3res     0zSi._eval_aseriescCr-rjrmr'r2r2r3rnur.zSi._eval_is_zero)rrrrr r4rr)r1rr2r3r6r5rrFrrnrr2r2rr3r,s F     r,csleZdZdZeZejZe ddZ e ddZ e ddZ e dd Z d d ZdddZfddZZS)r+a 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 cCtjSrz)rr)r<r2r2r3r2z Ci._atinfcCsttSrz)rrr<r2r2r3r3sz Ci._atneginfcCt|ttSrzr+rrrr2r2r3r6rBzCi._minusfactorcCt|ttd|Srrrrr@r2r2r3r5z Ci._IfactorcKs(ttt|ttt | dSr)rrrrr2r2r3rs(zCi._eval_rewrite_as_expintNrcCsj|jdj|||d}||d}|tjur%|j|dt|jr!dndd}|jr+tj S|j r3| |S|SNrrrrr) r&rrrrFrrrrKr rkr+rr2r2r3rs   zCi._eval_as_leading_termc sddlm}|d}|tjur^|jdfddtt|ddD|d||g}fddtt|ddD|d||g}tt|t t|St t | ||||S)NrrcrGrrHrrr2r3rrIz$Ci._eval_aseries..r6r%crJrrHrrr2r3rrK) rrrrGr&rrLr rrrr+rrMrrr3rs     (zCi._eval_aseriesr`)rrrrrr4rrr1rr2r3r6r5rrrrr2r2rr3r+{sO      r+c@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 cCrPrzrrGr<r2r2r3r2BrQz Shi._atinfcCrPrz)rrIr<r2r2r3r3FrQz Shi._atneginfcCr>rz)rrr2r2r3r6Jr=zShi._minusfactorcCr?rz)rr,r@r2r2r3r5NrBz Shi._IfactorcKs,t|tttt|dttdSr)rrrrrr2r2r3rRs,zShi._eval_rewrite_as_expintcCr-rjrmr'r2r2r3rnVr.zShi._eval_is_zeroNrcCsx|jd|}||d}|tjur"|j|dt|jrdndd}|jr'|S|j s/| |S|j r:t tj dS|S)Nrrrrr%) r&rrrrFrrrrKrr+rrPrr2r2r3r[s   zShi._eval_as_leading_termr`)rrrrrr4rr)r1rr2r3r6r5rrnrr2r2r2r3rs?    rc@s\eZdZdZeZejZe ddZ e ddZ e ddZ e dd Z d d ZdddZd S)ra  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 ========== .. [1] https://en.wikipedia.org/wiki/Trigonometric_integral cCrPrzrXr<r2r2r3r2rQz Chi._atinfcCrPrzrXr<r2r2r3r3rQz Chi._atneginfcCrRrzrUrr2r2r3r6rBzChi._minusfactorcCrTrrSr@r2r2r3r5rVz Chi._IfactorcKs.t tdt|tttt|dSr)rrrrrr2r2r3rrzChi._eval_rewrite_as_expintNrcCs|jdj|||d}||d}|tjur%|j|dt|jr!dndd}|jrD| |\}}|dur7t |n|}t |||tj S|j rL| |S|SrW)r&rrrrFrrrrKrrr rkr+)r,r/rrrTrr r r2r2r3rs   zChi._eval_as_leading_termr`)rrrrrr4rrr1rr2r3r6r5rrr2r2r2r3rksJ    rc@sJeZdZdZdZeddZdddZdd ZeZ d d Z d d Z e Z dS)FresnelIntegralz& Base class for the Fresnel integrals.TcCs|tjurtjS|jrtjStj}|}d}|d}|dur%| }|}d}|t}|dur9|jt|}|}d}|rA|||SdS)NFrT) rrGrtrKr)rHrOr_sign)rSrvprefactnewargchangedrr2r2r3rVs(    zFresnelIntegral.evalr6cCs0|dkr|tjt|jddSt||r)r4rrtrr&r r:r2r2r3r< s zFresnelIntegral.fdiffcCrfr`rgrcr2r2r3r riz&FresnelIntegral._eval_is_extended_realcCrfr`rmrcr2r2r3rn rizFresnelIntegral._eval_is_zerocCr_r`rarcr2r2r3rd rezFresnelIntegral._eval_conjugateNr)rrrrrrrVr<rrlrnrdr4r*r2r2r2r3rYs  rYcs^eZdZdZeZej Ze e ddZ ddZ ddZ dd Zdd d ZfddZZS)r}av Fresnel integral S. 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] http://dlmf.nist.gov/7 .. [3] http://mathworld.wolfram.com/FresnelIntegrals.html .. [4] http://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] http://dlmf.nist.gov/7 .. [3] http://mathworld.wolfram.com/FresnelIntegrals.html .. [4] http://functions.wolfram.com/GammaBetaErf/FresnelC .. [5] The converging factors for the fresnel integrals by John W. 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