g3dZddlmZddlmZddlmZddlmZm Z m Z ddl m Z m Z mZddlmZddlmZdd lmZdd lmZmZdd lmZdd lmZmZmZdd lmZm Z m!Z!ddl"m#Z#m$Z$ddl%m&Z&m'Z'ddl(m)Z)m*Z*m+Z+ddl,m-Z-m.Z.ddl/m0Z0m1Z1m2Z2ddl3m4Z4m5Z5dBdZ6GddeZ7GddeZ8GddeZ9GddeZ:GddeZ;Gd d!eZ<Gd"d#eZ=Gd$d%eZ>Gd&d'eZ?d(Z@Gd)d*eZAGd+d,eZBGd-d.eZCGd/d0eCZDGd1d2eCZEGd3d4eCZFGd5d6eCZGGd7d8eZHGd9d:eHZIGd;deZKGd?d@eZLdAS)Cz This module contains various functions that are special cases of incomplete gamma functions. It should probably be renamed. ) EulerGamma)Add)cacheit)FunctionArgumentIndexError expand_mul)IpiRational)is_eq)Pow)S)Dummyuniquely_named_symbol)sympify) factorial factorial2RisingFactorial) polar_liftre unpolarify)ceilingfloor)sqrtroot)explog exp_polar)coshsinh)cossinsinc)hypermeijergTc R|jdjr/|rd|d<|j|fi|tjfS|tjfS|r/|jdj|fi|\}}n"|jd\}}||t|zz||t|zz zdz }||t|zz||t|zz z dtzz }||fS)NrFcomplex)argsis_extended_realexpandrZero as_real_imagfuncr )selfdeephintsxyrims c/var/www/html/ai-engine/env/lib/python3.11/site-packages/sympy/functions/special/error_functions.pyreal_to_real_as_real_imagr6s" y|$"  "$E) DK....7 7!&> ! +"ty|"411511>>@@11y|((**1 ))A!G  tyyQqS11 11 4B ))A!G  tyyQqS11 1AaC 8B 8OceZdZdZdZddZddZedZe e dZ dZ d Z d Zd Zd Zd ZdZdZdZdZddZdZdZddZfdZeZxZS)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] https://dlmf.nist.gov/7 .. [3] https://mathworld.wolfram.com/Erf.html .. [4] https://functions.wolfram.com/GammaBetaErf/Erf Tc|dkr6dt|jddz zttz St ||Nr:r(rrr)rr rr/argindexs r5fdiffz erf.fdiffsF q==S$)A,/)***4883 3$T844 4r7ctSz8 Returns the inverse of this function. erfinvr>s r5inversez erf.inverses  r7c|jra|tjur tjS|tjur tjS|tjur tjS|jr tjSt|tr |j dSt|trtj|j dz S|jr tjSt|tr|j djr |j dS|t}|tjtjfvr|S|r ||  SdSNrr:) is_NumberrNaNInfinityOneNegativeInfinity NegativeOneis_zeror, isinstancerDr)erfcinverf2invextract_multiplicativelyr could_extract_minus_signclsargts r5evalzerf.evals> = ae||u  ""u ***}$ v c6 " " HQK  c7 # # '538A;& & ; 6M c7 # #  (; 8A;   ( ( + + Q/0 0 0J  ' ' ) ) CII:   r7cv|dks |dzdkr tjSt|}t|dz tdz }t |dkr|d |dzz|dz z||zz Sdtj|zz||zz|t |zttzz SNrr(r:) rr,rrlenrMrrr nr2previous_termsks r5 taylor_termzerf.taylor_terms q55AEQJJ6M Aq1uaddl##A>""Q&&&r**QT1QU;QqSAA))AqD0!IaLL.b2IJJr7cf||jdSNrr.r) conjugater/s r5_eval_conjugatezerf._eval_conjugate&yy1//11222r7c&|jdjSrcr)r*rfs r5 _eval_is_realzerf._eval_is_realy|,,r7cN|jdjrdS|jdjSNrT)r) is_finiter*rfs r5_eval_is_finitezerf._eval_is_finites' 9Q< ! 149Q<0 0r7c&|jdjSrcr)rNrfs r5 _eval_is_zerozerf._eval_is_zeroy|##r7c ddlm}t|dz|z tj|tj|dztt z z zSNr uppergammar('sympy.functions.special.gamma_functionsrxrrrKHalfr r/zkwargsrxs r5_eval_rewrite_as_uppergammazerf._eval_rewrite_as_uppergammasQFFFFFFAqDzz!|QUZZ1%=%=d2hh%FFGGr7c tjtz |zttz }tjtzt |tt |zz zSNrrKr rr fresnelcfresnelsr/r}r~rVs r5_eval_rewrite_as_fresnelszerf._eval_rewrite_as_fresnelsBuqy!mDHH$ HSMMAhsmmO;<>>>r7c t|dz|z |ttj|dzzttz z SNr(rexpintrr{r rs r5_eval_rewrite_as_expintzerf._eval_rewrite_as_expints;AqDzz!|aqvq!t 4 44T"XX===r7Nc ddlm}|rV|||tj}|tjur1tjt | t|dz zzStjt |t|dz zz S)Nr)limitr() sympy.series.limitsrrrJrLrM_erfsrrK)r/r}limitvarr~rlims r5_eval_rewrite_as_tractablezerf._eval_rewrite_as_tractables------  <%8QZ00Ca(((}uaRyyadU';;;uuQxxQTE ***r7c :tjt|z Sr)rrKerfcrs r5_eval_rewrite_as_erfczerf._eval_rewrite_as_erfcsutAwwr7c Bt tt|zzSrr erfirs r5_eval_rewrite_as_erfizerf._eval_rewrite_as_erfisr$qs))|r7rcN|jd|||}||d}|tjur ||d|dkrdnd}||jvr!|jrd|zttz S| |S)Nrlogxcdirr-+dirr() r)as_leading_termsubsrComplexInfinityr free_symbolsrNrr r.r/r2rrrVarg0s r5_eval_as_leading_termzerf._eval_as_leading_termsil**14d*CCxx1~~ 1$ $ $99Qdbjjssc9BBD  T\ S5b> !99T?? "r7c@ ddlm}|d}|tjtjfvr|jd |\}}n#ttf$r|cYSwxYw| }|j rt||z } fdt| D|d | zz |gz} tj t dz ttz t!| zz St#t$|||||S)NrOrderc~g|]9}tj|ztd|zdz zd|zdzzd|zzz :Sr(r:)rrMr.0r`r}s r5 z%erf._eval_aseries..s`+++]A% 1Q37(;(;;q1Q37|aQRd?RS+++r7r:r()sympy.series.orderrrrJrLr)leadterm ValueErrorNotImplementedError is_positiverrangerKrrr rsuperr9 _eval_aseries) r/r^args0r2rrpoint_exnewnsr} __class__s @r5rzerf._eval_aseriessD,,,,,,a QZ!34 4 4 ! A  1 22 34     B~ ?qt}}++++#Dkk+++.3eAagIq.A.A-BCuQTE 488 3sAw>>>S$--a4@@@sAA('A(r:rrc)__name__ __module__ __qualname____doc__ unbranchedr@rE classmethodrX staticmethodrrargrkrprsrrrrrrrrrrrr6r- __classcell__rs@r5r9r90sJJXJ5555[B  K K W\ K333---111 $$$HHH======NNN???>>>++++ # # # #AAAAA*-LLLLLr7r9ceZdZdZdZddZddZedZe e dZ dZ d Z dd Zd Zd ZdZdZdZdZdZdZdZddZeZdZd S)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 Tr:c|dkr6dt|jddz zttz St ||)Nr:r[rr(r=r>s r5r@z erfc.fdiff`sF q==c49Q<?*+++DHH4 4$T844 4r7ctSrBrPr>s r5rEz erfc.inversefs r7c|jrG|tjur tjS|tjur tjS|jr tjSt|trtj|j dz St|tr |j dS|jr tjS| t}|tjtj fvr| S|rd|| z SdS)Nrr()rHrrIrJr,rNrKrOrDr)rPrRr rLrSrTs r5rXz erfc.evalms = ae||u  ""v  u c6 " " '538A;& & c7 # # 8A;  ; 5L  ( ( + + Q/0 0 04K  ' ' ) ) !ssC4yy=  ! !r7c|dkr tjS|dks |dzdkr tjSt|}t |dz tdz }t |dkr|d |dzz|dz z||zz Sdtj|zz||zz|t|zttzz SrZ) rrKr,rrr\rMrrr r]s r5razerfc.taylor_terms 665L UUa!eqjj6M Aq1uaddl##A>""Q&&&r**QT1QU;QqSAA!-**QT11Yq\\>$r((3JKKr7cf||jdSrcrdrfs r5rgzerfc._eval_conjugaterhr7c&|jdjSrcrjrfs r5rkzerfc._eval_is_realrlr7Nc b|tdd|SN tractableT)r0rrewriter9r/r}rr~s r5rzerfc._eval_rewrite_as_tractable)||C  ((4((SSSr7c :tjt|z Sr)rrKr9rs r5_eval_rewrite_as_erfzerfc._eval_rewrite_as_erfsus1vv~r7c Ztjttt|zzzSr)rrKr rrs r5rzerfc._eval_rewrite_as_erfisuqac{""r7c tjtz |zttz }tjtjtzt |tt |zz zz Srrrs r5rzerfc._eval_rewrite_as_fresnelssIuqy!mDHH$u HSMMAhsmmO$CDDDr7c tjtz |zttz }tjtjtzt |tt |zz zz Srrrs r5rzerfc._eval_rewrite_as_fresnelcsIuQwk$r(("u HSMMAhsmmO$CDDDr7c tj|ttz t tjggdgt ddg|dzzz Sr)rrKrr r%r{r rs r5rzerfc._eval_rewrite_as_meijergsFuqbz'16(Bhr1oo=NPQSTPT"U"UUUUr7c tjd|zttz t tjgdtjzg|dz zz Sr)rrKrr r$r{rs r5rzerfc._eval_rewrite_as_hypers@uqs488|E16(QqvXJA$F$FFFFr7c ddlm}tjt |dz|z tj|tj|dzt t z z zz Srv)rzrxrrKrr{r r|s r5rz erfc._eval_rewrite_as_uppergammasXFFFFFFutAqDzz!|QUZZ1-E-Ed2hh-N%NOOOr7c tjt|dz|z z |ttj|dzztt z zSr)rrKrrr{r rs r5rzerfc._eval_rewrite_as_expintsButAqDzz!|#aqvq!t(<(<&>> 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 Tr:c|dkr5dt|jddzzttz St ||r<r=r>s r5r@z erfi.fdiffsC q==S1q)))$r((2 2$T844 4r7c|jrG|tjur tjS|jr tjS|tjur tjS|jr tjS|r ||  S|t}||tjurtSt|trt|j dzSt|tr"ttj |j dz zSt|tr)|j djrt|j dzSdSdSdSrG)rHrrIrNr,rJrSrRr rOrDr)rPrKrQrUr}nzs r5rXz erfi.eval"s? ; "AEzzu  "v ajz! 9 6M % % ' ' CGG8O ' ' * * >QZ"f%% $|#"g&& .!%"'!*,--"g&& $271:+= $|# > $ $ $ $r7cT|dks |dzdkr tjSt|}t|dz tdz }t |dkr|d|dzz|dz z||zz Sd||zz|t |zt tzz SrZ)rr,rrr\rrr r]s r5razerfi.taylor_term@s q55AEQJJ6M Aq1uaddl##A>""Q&&%b)AqD0AE:AaC@@1a4x9Q<<R!899r7cf||jdSrcrdrfs r5rgzerfi._eval_conjugateMrhr7c&|jdjSrcrjrfs r5_eval_is_extended_realzerfi._eval_is_extended_realPrlr7c&|jdjSrcrrrfs r5rszerfi._eval_is_zeroSrtr7Nc b|tdd|Srrrs r5rzerfi._eval_rewrite_as_tractableVrr7c Bt tt|zzSr)r r9rs r5rzerfi._eval_rewrite_as_erfYsr#ac(({r7c Pttt|zztz Sr)r rrs r5rzerfi._eval_rewrite_as_erfc\sac{Qr7c tjtz|zttz }tjtz t |tt |zz zSrrrs r5rzerfi._eval_rewrite_as_fresnels_rr7c tjtz|zttz }tjtz t |tt |zz zSrrrs r5rzerfi._eval_rewrite_as_fresnelccrr7c |ttz ttjggdgt ddg|dz zSrrrs r5rzerfi._eval_rewrite_as_meijerggs>bz'16(Bhr1oo5FANNNNr7c d|zttz ttjgdtjzg|dzzSrrrs r5rzerfi._eval_rewrite_as_hyperjs7s488|E16(QqvXJ1====r7c ddlm}t|dz |z |tj|dz tt z tjz zSrv)rzrxrrr{r rKr|s r5rz erfi._eval_rewrite_as_uppergammamsUFFFFFFQTE{{1}jj!Q$77R@15HIIr7c t|dz |z |ttj|dz zttz z Srrrs r5rzerfi._eval_rewrite_as_expintqs?QTE{{1}qA!6!66tBxx???r7c 6|tSrrrs r5rzerfi._eval_expand_functrr7rc*|jd|||}||d}||jvr!|jrd|zt t z S|jr||S||S)Nrrr() r)rrrrNrr ror.rs r5rzerfi._eval_as_leading_termysil**14d*CCxx1~~  T\ S5b> ! ^ #99T?? "yy~~r7cddlm}|d}|tjurv|jdfdt |D|d|zz |gz}t tdzttz t|zzStt| ||||S)Nrrc^g|])}td|zdz d|zd|zdzzzz *Sr)rrs r5rz&erfi._eval_aseries..sS'''AaC!G$$1q1Q37|(;<'''r7r:r()rrrrJr)rr rrr rrrr r/r^rr2rrrrr}rs @r5rzerfi._eval_aseriess,,,,,,a AJ   ! A''''"1XX'''*/%!Q$*:*:);>>JJJ@@@!!!-L B B B B B B B B Br7rc|eZdZdZdZedZdZdZdZ dZ dZ d Z d Z d Zd Zd ZdZdZdZdS)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/ c|j\}}|dkr+dt|dz zttz S|dkr+dt|dz zttz St ||)Nr:r[r()r)rrr rr/r?r2r3s r5r@z erf2.fdiffsqy1 q==c1a4%jj=b) ) ]]S!Q$ZZ<R( ($T844 4r7ctjtjtjf}|tjus|tjur tjS||kr tjS||vs||vrt |t |z St |tr|jd|kr |jdS|j s#|j s|j r|j s|j r&|j rt |t |z S| }| }|r|r|| |  S|s|rt |t |z SdSrG) rrJrLr,rIr9rOrQr)rNr* is_infiniterS)rUr2r3chksign_xsign_ys r5rXz erf2.evals]z1-qv6 ::ae5L !VV6M #XXcq66CFF? " a ! ! afQi1nn6!9  9 #  #Q%7 #AM #" #'(} #q66CFF? "++--++--  !v !CQBKK<  ! !q66#a&&=  ! !r7c||jd|jdSrGrdrfs r5rgzerf2._eval_conjugates:yy1//1149Q<3I3I3K3KLLLr7cJ|jdjo|jdjSrGrjrfs r5rzerf2._eval_is_extended_realsy|,N11NNr7c @t|t|z Srr9r/r2r3r~s r5rzerf2._eval_rewrite_as_erfs1vvAr7c @t|t|z Srrrs r5rzerf2._eval_rewrite_as_erfcsAwwa  r7c pttt|ztt|zz zSrrrs r5rzerf2._eval_rewrite_as_erfis&$qs))D1II%&&r7c t|tt|tz Sr)r9rrrs r5rzerf2._eval_rewrite_as_fresnels11vv~~h''#a&&..*B*BBBr7c t|tt|tz Sr)r9rrrs r5rzerf2._eval_rewrite_as_fresnelc r!r7c t|tt|tz Sr)r9rr%rs r5rzerf2._eval_rewrite_as_meijerg s11vv~~g&&Q)@)@@@r7c t|tt|tz Sr)r9rr$rs r5rzerf2._eval_rewrite_as_hypers11vv~~e$$s1vv~~e'<'<<&>tBxx&GGH AJJqL!%**QVQT":":488"CC DE Fr7c t|tt|tz Sr)r9rrrs r5rzerf2._eval_rewrite_as_expints11vv~~f%%Av(>(>>>r7c 6|tSrrrs r5rzerf2._eval_expand_funcrr7ct|jSr)r r)rfs r5rszerf2._eval_is_zerosdi  r7N)rrrrr@rrXrgrrrrrrrrrrrrsrr7r5rrs BBJ555!![!0MMMOOO!!!'''CCCCCCAAA===FFF ???!!!!!!!!r7rcDeZdZdZd dZd dZedZdZdZ dS) rDaR Inverse Error Function. The erfinv function is defined as: .. math :: \mathrm{erf}(x) = y \quad \Rightarrow \quad \mathrm{erfinv}(y) = x Examples ======== >>> from sympy import erfinv >>> from sympy.abc import x Several special values are known: >>> erfinv(0) 0 >>> erfinv(1) oo Differentiation with respect to $x$ is supported: >>> from sympy import diff >>> diff(erfinv(x), x) sqrt(pi)*exp(erfinv(x)**2)/2 We can numerically evaluate the inverse error function to arbitrary precision on [-1, 1]: >>> erfinv(0.2).evalf(30) 0.179143454621291692285822705344 See Also ======== erf: Gaussian error function. erfc: Complementary error function. erfi: Imaginary error function. erf2: Two-argument error function. erfcinv: Inverse Complementary error function. erf2inv: Inverse two-argument error function. References ========== .. [1] https://en.wikipedia.org/wiki/Error_function#Inverse_functions .. [2] https://functions.wolfram.com/GammaBetaErf/InverseErf/ r:c|dkrRttt||jddzzt jzSt||Nr:rr(rr rr.r)rr{rr>s r5r@z erfinv.fdiffTsR q==88C $)A, 7 7 :;;;AFB B$T844 4r7ctSrBrr>s r5rEzerfinv.inverseZs  r7c|tjur tjS|tjur tjS|jr tjS|tjur tjSt|tr|j dj r |j dS|jr tjS| d}|5t|tr"|j dj r|j d SdSdSdSNrr) rrIrMrLrNr,rKrJrOr9r)r*rRrs r5rXz erfinv.evalas ::5L !-  % % Y 6M !%ZZ:  a   !&)"< 6!9  9 6M ' ' + + >z"c22> 7T>GAJ;  >>>>>r7c &td|z SNr:rrs r5_eval_rewrite_as_erfcinvzerfinv._eval_rewrite_as_erfcinvwsacllr7c&|jdjSrcrrrfs r5rszerfinv._eval_is_zerozrtr7Nr) rrrrr@rErrXr2rsrr7r5rDrD!s//d5555 [*$$$$$r7rDcJeZdZdZd dZd dZedZdZdZ dZ d S) rPa Inverse Complementary Error Function. The erfcinv function is defined as: .. math :: \mathrm{erfc}(x) = y \quad \Rightarrow \quad \mathrm{erfcinv}(y) = x Examples ======== >>> from sympy import erfcinv >>> from sympy.abc import x Several special values are known: >>> erfcinv(1) 0 >>> erfcinv(0) oo Differentiation with respect to $x$ is supported: >>> from sympy import diff >>> diff(erfcinv(x), x) -sqrt(pi)*exp(erfcinv(x)**2)/2 See Also ======== erf: Gaussian error function. erfc: Complementary error function. erfi: Imaginary error function. erf2: Two-argument error function. erfinv: Inverse error function. erf2inv: Inverse two-argument error function. References ========== .. [1] https://en.wikipedia.org/wiki/Error_function#Inverse_functions .. [2] https://functions.wolfram.com/GammaBetaErf/InverseErfc/ r:c|dkrStt t||jddzzt jzSt||r+r,r>s r5r@z erfcinv.fdiffsT q==HH9S49Qs r5rEzerfcinv.inverses  r7c|tjur tjS|jr tjS|tjur tjS|dkr tjS|jr tjSdSr)rrIrNrJrKr,rLrUr}s r5rXz erfcinv.evalsf ::5L Y &:  !%ZZ6M !VV% % 9 :   r7c &td|z Sr1rCrs r5_eval_rewrite_as_erfinvzerfcinv._eval_rewrite_as_erfinvsac{{r7c,|jddz jSrGrrrfs r5rszerfcinv._eval_is_zeros ! q ))r7c&|jdjSrcrrrfs r5_eval_is_infinitezerfcinv._eval_is_infinitertr7Nr) rrrrr@rErrXr:rsr=rr7r5rPrP~s))X5555   [ ***$$$$$r7rPc4eZdZdZdZedZdZdS)rQa2 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/ c6|j\}}|dkr,t|||dz|dzz S|dkrHttt jzt|||dzzSt||)Nr:r()r)rr.rr rr{rrs r5r@z erf2inv.fdiffsy1 q==tyy1~~q(A-.. . ]]88AF?3tyy1~~q'8#9#99 9$T844 4r7c&|tjus|tjur tjS|jr|jr tjS|jr|tjur tjS|tjur|jr tjS|jrt |S|tjurt| S|jr|S|tjurt |S|jr"|jr tjSt |S|jr|SdSr)rrIrNr,rKrJrDrP)rUr2r3s r5rXz erf2inv.eval s ::ae5L Y 19 6M Y 1:::  !%ZZAIZ5L Y !99  !*__A2;;  Y H !*__!99  9 !y !v ayy 9 H  r7c>|j\}}|jr |jrdSdSdS)NTrr)r/r2r3s r5rszerf2inv._eval_is_zero(s9y1 9  4    r7N)rrrrr@rrXrsrr7r5rQrQsX00f555[4r7rQceZdZdZedZddZdZdZdZ dZ d Z e Z e Z e Zdd Zd Zdfd Zdfd ZfdZxZS)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 c6|jr tjS|tjur tjS|tjur tjS|jr tjS|\}}|r%t |dtztz|zzSdSr) rNrrLrJr,extract_branch_factorrCr r rUr}rr^s r5rXzEi.evals 9 % % !*__:  !$ $ $6M 9 &% %''))A  %b66AaCF1H$ $ % %r7r:ct|jd}|dkrt||z St||rG)rr)rrr/r?rVs r5r@zEi.fdiffs>1&& q==s88C< $T844 4r7c|jdtdz jr8tj||t t z|zStj||Sr/)r)rrr _eval_evalfr r r/precs r5rJzEi._eval_evalfs\ IaLB ' 4 O'd33qt6H6H6N6NN N#D$///r7c hddlm}|dtd|z ttzz S)Nrrwr)rzrxrr r r|s r5rzEi._eval_rewrite_as_uppergammas?FFFFFF 1jnnQ.///!B$66r7c dtdtd|z ttzz S)Nr:r)rrr r rs r5rzEi._eval_rewrite_as_expints)q*R..*+++ad22r7c t|trt|jdStt |Src)rOrlir)rrs r5_eval_rewrite_as_lizEi._eval_rewrite_as_lis: a   !afQi== #a&&zzr7c |jr/t|t|zttzz St|t|zSr) is_negativeShiChir r rs r5_eval_rewrite_as_SizEi._eval_rewrite_as_SisA = #q66CFF?QrT) )q66CFF? 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AAA F F F F F F111111 @ @ @ @ @ @ @ @ @r7rCcteZdZdZedZdZdZdZdZ dZ e Z e Z e Z d fd Zfd Zd ZxZS) 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 c ddlm}m}t|}||krt ||S|jr|dks|js> ! = MR1WWR]W"?PWjR!VZZB5J5J)JKKLL L&&((1 ;; F = T66"a==B$q&(1=26229R!V3D3DDZPQ]]UWZ[U[E\\] ]!Br ! $$q(!b1f+5eeAFmmCfRQRmmS Sr7c |j\}}|dkr%||dz z tgddgddd|z gg|zS|dkrt|dz | St||r+)r)r%rr)r/r?rxr}s r5r@z expint.fdifffsy A q==QK<QFQ1r6NB J JJ J ]]261%%% %$T844 4r7c >ddlm}||dz z|d|z |zS)Nrrwr:)rzrx)r/rxr}r~rxs r5rz"expint._eval_rewrite_as_uppergammaos7FFFFFF26{::a"fa0000r7c dkr>t|tt tzz ttzz Sjrdkrt | dz zt dz z t|tztt dz z tfdtdz DzzS|S)Nr:cFg|]}t|z dz |zzSr(rn)rr`rxr2s r5rz.expint._eval_rewrite_as_Ei..{s2HHHQiQ ++AqD0HHHr7) rCrr r rvrrE1rrrr)r/rxr}r~r2s ` @r5_eval_rewrite_as_Eizexpint._eval_rewrite_as_Eiss 77qA2b5)))***QrT1 1 ] rAvvAArAv;ya000Ar1B1BBAya(((HHHHH%Q--HHHIJJ JKr7c X|tjtfi|Sr)rrCrrs r5rzexpint._eval_expand_funcs)'t||B'88%888r7c P|dkr|St|t|z Sr1)rTrU)r/rxr}r~s r5rVzexpint._eval_rewrite_as_Sis& 77K1vvAr7rcb|jd|sl|jd}|dkr&|j|j}||||S|jr,|dkr&|j|j}||||St |||SrG)r)hasrVrirvrr)r/r2r^rrrxrkrs r5rizexpint._eval_nseriessy|"" 31BQww,D,di8q!T222 3266,D,di8q!T222ww$$Q4000r7cd ddlm}|d}|jd|tjurZ|jd  fdt |D|d |zz |gz}t  z t|zStt| ||||S)Nrrr:c\g|](}tj|zt|z|zz )Sr)rrMr)rr`rxr}s r5rz(expint._eval_aseries..s8TTTa!OB$:$::QTATTTr7) rrr)rrJrrrrrr) r/r^rr2rrrrrxr}rs @@r5rzexpint._eval_aseriess,,,,,,a Yq\ AJ   ! ATTTTT5QR88TTTX]X]^_`acd`d^dfgXhXhWiiAGGAIa( (VT""00E1dCCCr7cddlm}|j\}}tt d|j}||| zt | |zz|dtjfSNrr[rWr:) r]r\r)rrr^rrrJ)r/r)r~r\r^r2rWs r5raz expint._eval_rewrite_as_Integralsl666666y1 'T227 8 8xA2QBqD )Aq!*+=>>>r7ro)rrrrrrXr@rrrrVrprqrrrirrarrs@r5rrsddNTT[T*555111   999... 1 1 1 1 1 1 D D D D D???????r7rc"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. r:)r)r}s r5rrs@ !Q<<r7c~eZdZdZedZddZdZdZdZ dZ d Z e Z d Z e Zd Zd ZddZddZdZd S)rPa 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 c|jr tjS|tjur tjS|tjur tjS|jr tjSdSr)rNrr,rKrLrJr8s r5rXzli.eval/sU 9 6M !%ZZ% % !*__:  9 6M  r7r:c|jd}|dkrtjt|z St ||rGr)rrKrrrHs r5r@zli.fdiff:9il q==53s88# #$T844 4r7c||jd}|js'||SdSrc)r)is_extended_negativer.rer/r}s r5rgzli._eval_conjugateAs< IaL% ,99Q[[]]++ + , ,r7c @t|tdzSr)LirPrs r5_eval_rewrite_as_Lizli._eval_rewrite_as_LiG!uur!uu}r7c :tt|Sr)rCrrs r5rzli._eval_rewrite_as_EiJs#a&&zzr7c (ddlm}|dt|  tjtt|ttjt|z z zztt| z SNrrw)rzrxrrr{rKr|s r5rzli._eval_rewrite_as_uppergammaMsFFFFFFAAw'''CFF c!%A,&7&7789;>Aw<<H Ir7c ttt|ztttt|zzz tjttjt|z tt|z zz ttt|zz Sr)Cir rSirr{rKrs r5rVzli._eval_rewrite_as_SiRs1SVV8 qAc!ffH~-AE#a&&L))CAKK789;>qQx==I Jr7c tt|tt|z tjttjt|z tt|z zz Sr)rUrrTrr{rKrs r5rrzli._eval_rewrite_as_ShiXsWCFF c#a&&kk)AFCc!ff 4E4ECPQFF 4S,TTUr7c t|tddt|ztjtt|ttjt|z z zzt zS)N)r:r:)r(r()rr$rr{rKrrs r5rzli._eval_rewrite_as_hyper]seAuVVSVV444CFF c!%A,&7&7789;EF Gr7c &tt|  tjttjt|z tt|z zz t ddt| z S)N)rr))rrr)rrr{rKr%rs r5rzli._eval_rewrite_as_meijergasgc!ffW AE#a&&L(9(9CAKK(G HH*lSVVG<<= >r7Nc @|tt|zSr)rXrrs r5rzli._eval_rewrite_as_tractablees4A<<r7rc|jdfdtd|D}tttzt |zS)NrcZg|]'}t|zt||zz (Sr)rrrs r5rz$li._eval_nseries..js3 C C C!c!ffq[IaLL1, - C C Cr7r:)r)rrrr)r/r2r^rrrr}s @r5rizli._eval_nserieshsO IaL C C C CuQ{{ C C CCAKK'#q'11r7c2|jd}|jrdSdSrnrrrs r5rszli._eval_is_zerom& IaL 9 4  r7rrro)rrrrrrXr@rgrrrrVrprrrqrrrrirsrr7r5rPrPs``F[5555,,, III JJJ.VVV0GGG>>>    2222 r7rPcLeZdZdZedZd dZdZdZd dZ d d Z dS)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 cx|tjur tjS|tdkr tjSdSr)rrJr,r8s r5rXzLi.evals2  ??:  !A$$YY6MYr7r:c|jd}|dkrtjt|z St ||rGrrHs r5r@zLi.fdiffrr7c\|t|Sr)rrPevalfrKs r5rJzLi._eval_evalfs"||B%%d+++r7c @t|tdz Sr)rPrs r5rQzLi._eval_rewrite_as_lirr7Nc `|tddS)NrT)r0)rrPrs r5rzLi._eval_rewrite_as_tractables'||B'' $'???r7rcN|j|j}||||Sr)rQr)ri)r/r2r^rrrks r5rizLi._eval_nseriess) $D $di 0q!T***r7rrro) rrrrrrXr@rJrQrrirr7r5rrrs==@[ 5555,,,@@@@++++++r7rcNeZdZdZedZd dZdZdZd fd Z xZ S) TrigonometricIntegralz) Base class for trigonometric integrals. c|tjur|jS|tjur|S|tjur|S|jr|jS|tt}|3| ddkr|t}|| |dS|tt }|| |dS|td}|.| ddkr|d}|| |S|\}}|dkr||krdSdtztz|z| dz||zS)Nrr:rr()rr,_atzerorJ_atinfrL _atneginfrNrRrr _trigfunc_Ifactor _minusfactorrEr rFs r5rXzTrigonometricIntegral.evals ;;;  !*__::<<  !$ $ $==?? " 9 ;   ' ' 1 6 6 :#--**a//++A..B ><<A&& &  ' ' A2 7 7 ><<B'' '  ' ' 2 7 7 :#--**a//++B//B >##B'' '''))A 66bAgg FtAvax a(((33r7722r7r:ct|jd}|dkr|||z St||rG)rr)rrrHs r5r@zTrigonometricIntegral.fdiffsE1&& q==>>#&&s* *$T844 4r7c \||tSr)rrrCrs r5rz)TrigonometricIntegral._eval_rewrite_as_Eis$++A..66r:::r7c ^ddlm}|||Sr)rzrxrrr|s r5rz1TrigonometricIntegral._eval_rewrite_as_uppergammas6FFFFFF++A..66zBBBr7rc2|jd|ddkr#t|||S|||||}|ddkr|dz}|t dd}|ddkr|tt|zz }|||jd|||S)Nrr:c||z|z Srr)rWr^s r5z5TrigonometricIntegral._eval_nseries.. s!Q$q&r7F) simultaneous) r)rrrirreplacer rr)r/r2r^rr baseseriesrs r5riz#TrigonometricIntegral._eval_nseriess 9Q<  Q " "a ' '77((At44 4^^A&&44Q4@@ >>!   ! ! !OJ''-@-@u'UU >>!   ! ! *s1vv- -Jq$)A,//==aDIIIr7rro) rrrrrrXr@rrrirrs@r5rrs3333[3>5555;;;CCC J J J J J J J J J Jr7rceZdZdZeZejZe dZ e dZ e dZ e dZ dZdZeZd d Zfd Zd ZxZS)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 c*ttjzSrr rr{rUs r5rz Si._atinf[s!&yr7c,t tjzSrrrs r5rz Si._atneginf_ss16zr7c"t| Sr)rr8s r5rzSi._minusfactorcs1v r7c6tt|z|zSr)r rTrUr}signs r5rz Si._IfactorgsQx}r7c tdz ttt|zttt |zz dz tz zSr)r rrr rs r5rzSi._eval_rewrite_as_expintksG!tr*Q--/**R A2q0@-A-AA1DQFFFr7c ddlm}ttd|gj}|t ||d|fSrZ)r]r\rrr^r#r`s r5razSi._eval_rewrite_as_IntegralosO666666 'aS116 7 7xQ!Q+++r7Nrc:|jd|||}||d}|tjur.||dt |jrdnd}|jr|S|j s| |S|SNrrrrr r)rrrrIrrrSrNrr.rs r5rzSi._eval_as_leading_termvsil**14d*CCxx1~~ 15==99Qbhh.B'Kss9LLD < J! 99T?? "Kr7c ddlm}|d}|tjur|jd fdt |dzdzD|d |zz |gz} fdt |dzD|d |zz |gz}t dz t t|zz t t|zz Stt| ||||S)Nrrclg|]0}tj|ztd|zzd|zdzzz 1SrrrMrrs r5rz$Si._eval_aseries..P...!IacNN2Q1q\A...r7r(r:crg|]3}tj|ztd|zdzzd|dzzzz 4Srrrs r5rz$Si._eval_aseries..V***!IacAg$6$66QAYG***r7) rrrrJr)rr r!rr"rrr) r/r^rr2rrrpqr}rs @r5rzSi._eval_aseriess4,,,,,,a AJ   ! A...."1a4!8__...16qAvq1A1A0BCA****"1a4[[***-2U1QT61-=-=,>?Aa4#a&&a.(3q66#q'>9 9R,,Qq$???r7c2|jd}|jrdSdSrnrrrs r5rszSi._eval_is_zerorr7rc)rrrrr"rrr,rrrrrrrra_eval_rewrite_as_sincrrrsrrs@r5rrsDDLIfG[[[[GGG,,, 7    @@@@@ r7rceZdZdZeZejZe dZ e dZ e dZ e dZ dZdZd d Zfd ZxZS) 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 ctjSr)rr,rs r5rz Ci._atinfs v r7c ttzSr)r r rs r5rz Ci._atneginfs t r7c@t|ttzzSrrr r r8s r5rzCi._minusfactors!uuqt|r7cLt|ttzdz |zzSrrUr r rs r5rz Ci._Ifactors1vv"Qt ##r7c ttt|zttt |zz dz Sr)rrr rs r5rzCi._eval_rewrite_as_expints<JqMM!O$$r*aR..*:';';;>r7c ddlm}ttd|gj}t jt|z|dt|z |z |d|fz Sr) r]r\rrr^rrrr!r`s r5razCi._eval_rewrite_as_Integralsi666666 'aS116 7 7|c!ff$xx3q661 q!Qi'H'HHHr7Nrc|jd|||}||d}|tjur.||dt |jrdnd}|jrH| |\}}|t|n|}t|||zztzS|j r| |S|Srr)rrrrIrrrSrNrdrrror.r/r2rrrVrrfrgs r5rzCi._eval_as_leading_termil**14d*CCxx1~~ 15==99Qbhh.B'Kss9LLD < ((++DAq!\3q666tDq66AdF?Z/ / ^ 99T?? "Kr7c> ddlm}|d}|tjtjfvr|jd fdt |dzdzD|d |zz |gz} fdt |dzD|d |zz |gz}t t|zt t|zz } |tjur| ttzz } | Stt|||||S)Nrrclg|]0}tj|ztd|zzd|zdzzz 1Srrrs r5rz$Ci._eval_aseries..rr7r(r:crg|]3}tj|ztd|zdzzd|dzzzz 4Srrrs r5rz$Ci._eval_aseries..rr7)rrrrJrLr)rr"rr!r r rrr) r/r^rr2rrrrrresultr}rs @r5rzCi._eval_aseriessP,,,,,,a QZ!34 4 4 ! A...."1a4!8__...16qAvq1A1A0BCA****"1a4[[***-2U1QT61-=-=,>?AVVS!W%AQ(88F***!B$MR,,Qq$???r7rc)rrrrr!rrrrrrrrrrrarrrrs@r5rrsMM^IG[[[$$[$???III    @@@@@@@@@r7rceZdZdZeZejZe dZ e dZ e dZ e dZ dZdZd d ZdS) rTa 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 ctjSrrrJrs r5rz Shi._atinfh zr7ctjSr)rrLrs r5rz Shi._atneginfls !!r7c"t| Sr)rTr8s r5rzShi._minusfactorpsAwr7c6tt|z|zSr)r rrs r5rz Shi._IfactortsAwt|r7c t|ttttz|zz dz ttzdz z Sr)rrr r rs r5rzShi._eval_rewrite_as_expintxs<19QrT??1,---q01R4699r7c2|jd}|jrdSdSrnrrrs r5rszShi._eval_is_zero|rr7Nrc4|jd|}||d}|tjur.||dt |jrdnd}|jr|S|j s| |S|S)Nrrrrrrs r5rzShi._eval_as_leading_termsil**1--xx1~~ 15==99Qbhh.B'Kss9LLD < J! 99T?? "Kr7rc)rrrrr rrr,rrrrrrrrsrrr7r5rTrT%s==~IfG[""["[[:::      r7rTceZdZdZeZejZe dZ e dZ e dZ e dZ dZd d ZdS) rUa  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 ctjSrrrs r5rz Chi._atinfrr7ctjSrrrs r5rz Chi._atneginfrr7c@t|ttzzSrrr8s r5rzChi._minusfactors1vv"}r7cLt|ttzdz |zzSrrrs r5rz Chi._Ifactors!uuqtAvd{""r7c t tzdz t|ttttz|zzdz z Sr)r r rrrs r5rzChi._eval_rewrite_as_expints>r"uQw"Q%%"Yqt__Q%6"7"77:::r7Nrc|jd|||}||d}|tjur.||dt |jrdnd}|jrH| |\}}|t|n|}t|||zztzS|j r| |S|Srrrs r5rzChi._eval_as_leading_termrr7rc)rrrrrrrrrrrrrrrrrr7r5rUrUsHHTIG[[[##[#;;;      r7rUcNeZdZdZdZedZd dZdZeZ dZ dZ e Z d S) FresnelIntegralz& Base class for the Fresnel integrals.Tc@|tjur tjS|jr tjStj}|}d}|d}|| }|}d}|t}||jtz|z}|}d}|r|||zSdS)NFrT) rrJr{rNr,rKrRr _sign)rUr}prefactnewargchangedrs r5rXzFresnelIntegral.eval s  ??6M 9 6M%  , ,R 0 0 >hGFG  , ,Q / / >ik')GFG  '33v;;& & ' 'r7r:c|dkr8|tjtz|jddzzSt ||r+)rrr{r r)rr>s r5r@zFresnelIntegral.fdiff( sB q==>>!&)DIaL!O";<< <$T844 4r7c&|jdjSrcrjrfs r5rz&FresnelIntegral._eval_is_extended_real. rlr7c&|jdjSrcrrrfs r5rszFresnelIntegral._eval_is_zero3 rtr7cf||jdSrcrdrfs r5rgzFresnelIntegral._eval_conjugate6 rhr7Nr)rrrrrrrXr@rrprsrgr6r-rr7r5rr s00J''['<5555 ----O$$$333-LLLr7rc|eZdZdZeZej Ze e dZ dZ dZ dZdZd d Zfd ZxZS) ray 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] 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. Wrench Jr. and Vicki Alley c|dkr tjSt|}t|dkr=|d}tdz |dzzd|zdz zd|zd|zdzzd|zdzzz |zS|dz|dz |zztdd|zdz ztd|zdzzzzd|zdzt d|zdzzz S) Nrr:rr(rr[rr,rr\r rr^r2r_rs r5razfresnels.taylor_term s q556M A>""Q&&"2&Qq!t QqS1W-qsAaC!G}acAg/FG1LL!t1uqj(AaDD2a4!8,. s6661q1!IacAg$6$66acAg,QqS1W-AaC81Q3GI)4r7r:r(cg|]m}d|zdzktj|ztd|zdz zdd|zdzzd|zdzzzdd|zdz zztd|zdz zz nSrr:r(rrs r5rz*fresnels._eval_aseries.. 6661q1]A- !A#'0B0BBacAg,QqS1W-AaC!G . (***1$qt**Q***r7cDg|]}tdtz  |zSr~r r!s r5rz*fresnels._eval_aseries.. r"r7r)rrrrJr)rr{r"rr!rrr rr r/r^rr2rrrrrrr}rs ` @r5rzfresnels._eval_aseries s,,,,,,a QZ!*- - - ! A666661++666AAaC 666661++6666A+****A*****Aaj((bAQV8s1a4yya03q!t99S!W3DD$q$qt**Q,''(*/%!Q$*:*:; ;ww$$Qq$777r7rc)rrrrr"rrrKrrrrarrrrarrrrs@r5rr< sSShI UFE  m m W\ maaaeee[[[333 # # # #888888888r7rczeZdZdZeZejZe e dZ dZ dZ dZdZd d Zfd ZxZS) rau Fresnel integral C. 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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