oVh6dZddlmZddlmZddlmZddlmZm Z m Z ddl m Z ddl mZmZmZmZddlmZdd lmZdd lmZdd lmZmZdd lmZdd lmZmZm Z ddl!m"Z"m#Z#m$Z$ddl%m&Z&m'Z'ddl(m)Z)m*Z*ddl+m,Z,m-Z-m.Z.ddl/m0Z0m1Z1ddl2m3Z3m4Z4m5Z5ddl6m7Z7m8Z8dBdZ9GddeZ:GddeZ;GddeZ<GddeZ=GddeZ>Gd d!eZ?Gd"d#eZ@Gd$d%eZAGd&d'eZBd(ZCGd)d*eZDGd+d,eZEGd-d.eZFGd/d0eFZGGd1d2eFZHGd3d4eFZIGd5d6eFZJGd7d8eZKGd9d:eKZLGd;deZNGd?d@eZOyA)Cz This module contains various functions that are special cases of incomplete gamma functions. It should probably be renamed. ) EulerGamma)Add)cacheit)DefinedFunctionArgumentIndexError expand_mul)fuzzy_or)IpiRationalInteger)is_eq)Pow)S)Dummyuniquely_named_symbol)sympify) factorial factorial2RisingFactorial) polar_liftre unpolarify)ceilingfloor)sqrtroot)explog exp_polar)coshsinh)cossinsinc)hypermeijergc R|jdjr<|r(d|d<|j|fi|tjfS|tjfS|r2|jdj|fi|j \}}n |jdj \}}|j |t|zz|j |t|zz zdz }|j |t|zz|j |t|zz z dtzz }||fS)NrFcomplex)argsis_extended_realexpandrZero as_real_imagfuncr )selfdeephintsxyrims W/home/dcms/DCMS/lib/python3.12/site-packages/sympy/functions/special/error_functions.pyreal_to_real_as_real_imagr8s yy|$$ $E) DKK..7 7!&&> ! "tyy|""4151>>@1yy|((*1 ))A!G tyyQqS1 11 4B ))A!G tyyQqS1 1AaC 8B 8OceZdZdZdZddZddZedZe e dZ dZ dZ d Zd Zd Zd Zd ZdZdZdZdZdZdZddZdZdZdZfdZeZxZ S)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|dk(r/dt|jddz zttz St ||Nr*rrr+rr rr1argindexs r7fdiffz erf.fdiffs> q=S$))A,/)**483 3$T84 4r9ctSz8 Returns the inverse of this function. erfinvr@s r7inversez erf.inverses  r9c |jr|tjurtjS|tjurtjS|tj urtj S|jrtjSt|tr|jdSt|tr tj|jdz S|jrtjSt|tr(|jdjr|jdS|jt}|tjtj fvr|S|j!r ||  SyNrr>) is_NumberrNaNInfinityOneNegativeInfinity NegativeOneis_zeror. isinstancerFr+erfcinverf2invextract_multiplicativelyr could_extract_minus_signclsargts r7evalzerf.evals ==aee|uu  "uu ***}}$vv c6 "88A;  c7 #55388A;& & ;;66M c7 # (;(;88A;   ( ( + Q//0 0J  ' ' )I:  *r9cH|dks|dzdk(rtjSt|}t|dz tdz }t |dkDr|d |dzz|dz z||zz Sdtj |zz||zz|t |zttzz SNrr*r>) rr.rrlenrOrrr nr4previous_termsks r7 taylor_termzerf.taylor_terms q5AEQJ66M Aq1uadl#A>"Q&&r**QT1QU;QqSAA))AqD0!IaL.b2IJJr9cZ|j|jdjSNrr0r+ conjugater1s r7_eval_conjugatezerf._eval_conjugate"yy1//122r9c<|jdjduryyNrTr+r,rhs r7 _eval_is_realzerf._eval_is_reals 99Q< ( (D 0 1r9c<|jdjduryyrl)r+ is_imaginaryrhs r7_eval_is_imaginaryzerf._eval_is_imaginarys 99Q< $ $ , -r9cb|jd}t|j|jgSre)r+r is_finiter,r1zs r7_eval_is_finitezerf._eval_is_finites) IIaLa&8&89::r9cl|jdjdur|jdjSyrl)r+r,rPrhs r7 _eval_is_zerozerf._eval_is_zeros1 99Q< ( (D 099Q<'' ' 1r9cl|jdjdur|jdjSyrl)r+r,is_extended_positiverhs r7_eval_is_positivezerf._eval_is_positive1 99Q< ( (D 099Q<44 4 1r9cl|jdjdur|jdjSyrl)r+r,is_extended_negativerhs r7_eval_is_negativezerf._eval_is_negativer|r9c ddlm}t|dz|z tj|tj |dztt z z zSNr uppergammar*'sympy.functions.special.gamma_functionsrrrrMHalfr r1rukwargsrs r7_eval_rewrite_as_uppergammazerf._eval_rewrite_as_uppergammas=FAqDz!|QUUZ1%=d2h%FFGGr9c tjtz |zttz }tjtzt |tt |zz zSNrrMr rr fresnelcfresnelsr1rurrXs r7_eval_rewrite_as_fresnelszerf._eval_rewrite_as_fresnels@uuqy!mDH$ HSMAhsmO;<>>r9c t|dz|z |ttj|dzzttz z SNr*rexpintrrr rs r7_eval_rewrite_as_expintzerf._eval_rewrite_as_expints6AqDz!|aqvvq!t 44T"X===r9c ddlm}|rW|||tj}|tjur-tj t | t|dz zzStjt |t|dz zz S)Nr)limitr*) sympy.series.limitsrrrLrNrO_erfsrrM)r1rulimitvarrrlims r7_eval_rewrite_as_tractablezerf._eval_rewrite_as_tractablesn- 8QZZ0Ca(((}}uaRyadU';;;uuuQxQTE ***r9c :tjt|z Sr)rrMerfcrs r7_eval_rewrite_as_erfczerf._eval_rewrite_as_erfcsuutAwr9c 6t tt|zzSrr erfirs r7_eval_rewrite_as_erfizerf._eval_rewrite_as_erfisr$qs)|r9cB|jdj|||}|j|d}|tjur|j |d|dk(rdnd}||j vr!|jrd|zttz S|j|S)Nrlogxcdirr-+dirr*) r+as_leading_termsubsrComplexInfinityr free_symbolsrPrr r0r1r4rrrXarg0s r7_eval_as_leading_termzerf._eval_as_leading_termsiil**14d*Cxx1~ 1$$ $99Qdbjsc9BD   T\\S5b> !99T? "r9ctddlm}|d}|tjtjfvr|j d} |j |\}} | } | jrt|| z } t| D cgc]9} tj| ztd| zdz z|d| zdzzd| zzz ;c} |d|| zz |gz} tjt|dz t!t"z t%| zz St&t(|W||||S#ttf$r|cYSwxYwcc} wNrOrderr*r>)sympy.series.orderrrrLrNr+leadterm ValueErrorNotImplementedError is_positiverrangerOrrMrrr rsuperr; _eval_aseries)r1r`args0r4rrpointru_exnewnrbs __class__s r7rzerf._eval_aseriess6,a QZZ!3!34 4 ! A  1 2B~~qt}#Dk+]]A% 1Q37(;;q1Q37|aQRd?RS+.3AagIq.A-BCuuQTE 48 3sAw>>>S$-a4@@ 34    +sD=>D5D21D2r>r)!__name__ __module__ __qualname____doc__ unbranchedrBrG classmethodrZ staticmethodrrcrirnrqrvrxr{rrrrrrrrrrrrr8r/ __classcell__rs@r7r;r;1sJXJ5B  K  K3 ;(55H==N?>+ #A*-Lr9r;ceZdZdZdZddZddZedZe e dZ dZ dZ dd Zd Zd Zd ZdZdZdZdZdZdZdZeZdZy )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 Tc|dk(r/dt|jddz zttz St ||)Nr>r]rr*r?r@s r7rBz erfc.fdiffos> q=c499Q<?*++DH4 4$T84 4r9ctSrDrRr@s r7rGz erfc.inverseus r9c^|jr`|tjurtjS|tjurtjS|j rtj St|tr tj |jdz St|tr|jdS|j rtj S|jt}|tjtjfvr| S|jr d|| z SyNrr*)rJrrKrLr.rPrMrQrFr+rRrTr rNrUrVs r7rZz erfc.eval|s ==aee|uu  "vv uu c6 "55388A;& & c7 #88A;  ;;55L  ( ( + Q//0 04K  ' ' )sC4y=  *r9cr|dk(rtjS|dks|dzdk(rtjSt|}t |dz tdz }t |dkDr|d |dzz|dz z||zz Sdtj |zz||zz|t|zttzz Sr\) rrMr.rrr^rOrrr r_s r7rczerfc.taylor_terms 655L Ua!eqj66M Aq1uadl#A>"Q&&r**QT1QU;QqSAA!--**QT11Yq\>$r(3JKKr9cZ|j|jdjSrerfrhs r7rizerfc._eval_conjugaterjr9ct|jdjdury|jdjduryy)NrTF)r+r,rprhs r7rnzerfc._eval_is_reals9 99Q< ( (D 0 99Q< $ $ , -r9Nc P|jtjdd|SN tractableT)r2rrewriter;r1rurrs r7rzerfc._eval_rewrite_as_tractable#||C ((4((SSr9c :tjt|z Sr)rrMr;rs r7_eval_rewrite_as_erfzerfc._eval_rewrite_as_erfsuus1v~r9c Vtjttt|zzzSr)rrMr rrs r7rzerfc._eval_rewrite_as_erfisuuqac{""r9c tjtz |zttz }tjtjtzt |tt |zz zz Srrrs r7rzerfc._eval_rewrite_as_fresnelssIuuqy!mDH$uu HSMAhsmO$CDDDr9c tjtz |zttz }tjtjtzt |tt |zz zz Srrrs r7rzerfc._eval_rewrite_as_fresnelcsIuuQwk$r("uu HSMAhsmO$CDDDr9c tj|ttz t tj ggdgt ddg|dzzz Sr)rrMrr r'rr rs r7rzerfc._eval_rewrite_as_meijergsCuuqbz'166(Bhr1o=NPQSTPT"UUUUr9c tjd|zttz t tj gdtj zg|dz zz Sr)rrMrr r&rrs r7rzerfc._eval_rewrite_as_hypersAuuqs48|E166(QqvvXJA$FFFFr9c ddlm}tjt |dz|z tj|tj |dzt t z z zz Sr)rrrrMrrr rs r7rz erfc._eval_rewrite_as_uppergammasFFuutAqDz!|QUUZ1-Ed2h-N%NOOOr9c tjt|dz|z z |ttj|dzztt z zSr)rrMrrrr rs r7rzerfc._eval_rewrite_as_expints?uutAqDz!|#aqvvq!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 Tc|dk(r.dt|jddzzttz St ||r=r?r@s r7rBz erfi.fdiff.s; q=S1q))$r(2 2$T84 4r9c|jr`|tjurtjS|jrtjS|tj urtj S|jrtjS|j r ||  S|jt}||tj urtSt|trt|jdzSt|tr'ttj|jdz zSt|tr0|jdjrt|jdzSyyyrI)rJrrKrPr.rLrUrTr rQrFr+rRrMrSrWrunzs r7rZz erfi.eval4s ;;AEEzuu vv ajjzz! 9966M % % 'G8O ' ' * >QZZ"f%|#"g&!%%"''!*,--"g&2771:+=+=|#,>& r9c|dks|dzdk(rtjSt|}t|dz tdz }t |dkDr|d|dzz|dz z||zz Sd||zz|t |zt tzz Sr\)rr.rrr^rrr r_s r7rczerfi.taylor_termRs q5AEQJ66M Aq1uadl#A>"Q&%b)AqD0AE:AaC@@1a4x9Q<R!899r9cZ|j|jdjSrerfrhs r7rizerfi._eval_conjugate_rjr9c4|jdjSrermrhs r7_eval_is_extended_realzerfi._eval_is_extended_realbyy|,,,r9c4|jdjSrer+rPrhs r7rxzerfi._eval_is_zeroeyy|###r9c P|jtjdd|Srrrs r7rzerfi._eval_rewrite_as_tractablehrr9c 6t tt|zzSr)r r;rs r7rzerfi._eval_rewrite_as_erfksr#ac({r9c Bttt|zztz Sr)r rrs r7rzerfi._eval_rewrite_as_erfcnsac{Qr9c tjtz|zttz }tjtz t |tt |zz zSrrrs r7rzerfi._eval_rewrite_as_fresnelsqrr9c tjtz|zttz }tjtz t |tt |zz zSrrrs r7rzerfi._eval_rewrite_as_fresnelcurr9c |ttz ttjggdgt ddg|dz zSrrrs r7rzerfi._eval_rewrite_as_meijergys9bz'166(Bhr1o5FANNNr9c d|zttz ttjgdtjzg|dzzSrrrs r7rzerfi._eval_rewrite_as_hyper|s6s48|E166(QqvvXJ1===r9c ddlm}t|dz |z |tj|dz tt z tj z zSr)rrrrrr rMrs r7rz erfi._eval_rewrite_as_uppergammasAFQTE{1}j!Q$7R@155HIIr9c t|dz |z |ttj|dz zttz z Srrrs r7rzerfi._eval_rewrite_as_expints:QTE{1}qA!66tBx???r9c ,|jtSrrrs r7rzerfi._eval_expand_funcrr9c"|jdj|||}|j|d}||jvr!|jrd|zt t z S|jr|j|S|j|S)Nrrr*) r+rrrrPrr rsr0rs r7rzerfi._eval_as_leading_termsyiil**14d*Cxx1~   T\\S5b> ! ^^99T? "yy~r9cddlm}|d}|tjur|jd}t |Dcgc]%}t d|zdz d|z|d|zdzzzz 'c}|d||zz |gz} t t|dzttz t| zzStt|;||||Scc}wr)rrrrLr+rrr rrr rrrr r1r`rr4rrrrurbrrs r7rzerfi._eval_aseriess,a AJJ  ! A"1X'AaC!G$1q1Q37|(;<'*/!Q$*:);rr)rrrrrrBrrZrrrcrirrxrrrrrrrrrrr8r/rrrrs@r7rrsGRJ5 $$:  :  :3-$T==O>J@!-L B Br9rcteZdZdZdZedZdZdZdZ dZ dZ d Z d Z d Zd Zd ZdZdZdZy)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\}}|dk(r"dt|dz zttz S|dk(r"dt|dz zttz St ||)Nr>r]r*)r+rrr rr1rAr4r5s r7rBz erf2.fdiffsdyy1 q=c1a4%j=b) ) ]S!Q$Z<R( ($T84 4r9ctjtjtjf}|tjus|tjurtjS||k(rtjS||vs||vrt |t |z St |tr!|jd|k(r|jdS|js<|js0|jr |js|jr#|jrt |t |z S|j}|j}|r|r || |  S|s|rt |t |z SyrI) rrLrNr.rKr;rQrSr+rPr, is_infiniterU)rWr4r5chksign_xsign_ys r7rZz erf2.evalszz1--qvv6 :aee55L !V66M #Xcq6CF? " a !affQi1n66!9  99 Q%7%7AMM""q}}q6CF? "++-++- vQBK< q6#a&= r9c|j|jdj|jdjSrIrfrhs r7rizerf2._eval_conjugate s5yy1//1499Q<3I3I3KLLr9cj|jdjxr|jdjSrIrmrhs r7rzerf2._eval_is_extended_real s)yy|,,N11N1NNr9c 0t|t|z Srr;r1r4r5rs r7rzerf2._eval_rewrite_as_erfs1vAr9c 0t|t|z Srrr%s r7rzerf2._eval_rewrite_as_erfcsAwa  r9c Zttt|ztt|zz zSrrr%s r7rzerf2._eval_rewrite_as_erfis"$qs)D1I%&&r9c |t|jtt|jtz Sr)r;rrr%s r7rzerf2._eval_rewrite_as_fresnels'1v~~h'#a&..*BBBr9c |t|jtt|jtz Sr)r;rrr%s r7rzerf2._eval_rewrite_as_fresnelcr*r9c |t|jtt|jtz Sr)r;rr'r%s r7rzerf2._eval_rewrite_as_meijergs'1v~~g&Q)@@@r9c |t|jtt|jtz Sr)r;rr&r%s r7rzerf2._eval_rewrite_as_hyper"s'1v~~e$s1v~~e'<<tBx&GGH AJqL!%%*QVVQT":48"CC DE Fr9c |t|jtt|jtz Sr)r;rrr%s r7rzerf2._eval_rewrite_as_expint*s'1v~~f%Av(>>>r9c ,|jtSrrrs r7rzerf2._eval_expand_func-rr9c&t|jSr)rr+rhs r7rxzerf2._eval_is_zero0sdii  r9N)rrrrrBrrZrirrrrrrrrrrrrxrr9r7rrsiBJ5!!0MO!'CCA=F ?!!r9rc<eZdZdZddZddZedZdZdZ y) rFaR 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/ c|dk(rKttt|j|jddzzt j zSt||Nr>rr*rr rr0r+rrrr@s r7rBz erfinv.fdifffsI q=8C $))A, 7 :;;AFFB B$T84 4r9ctSrDr$r@s r7rGzerfinv.inversels  r9cL|tjurtjS|tjurtjS|jrtj S|tj urtjSt|tr(|jdjr|jdS|jrtj S|jd}|;t|tr*|jdjr|jd SyyyNrr) rrKrOrNrPr.rMrLrQr;r+r,rTrs r7rZz erfinv.evalss :55L !-- %% % YY66M !%%Z::  a !&&)"<"<66!9  9966M ' ' + >z"c2 7T7TGGAJ; 8U2>r9c td|z SNr>rrs r7_eval_rewrite_as_erfcinvzerfinv._eval_rewrite_as_erfcinvsqs|r9c4|jdjSrer rhs r7rxzerfinv._eval_is_zeror r9Nr) rrrrrBrGrrZr;rxrr9r7rFrF3s0/d5 *$r9rFcBeZdZdZd dZd dZedZdZdZ dZ y) rRa 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/ c|dk(rLtt t|j|jddzzt j zSt||r4r5r@s r7rBz erfcinv.fdiffsK q=H9S499Q>> 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/ c|j\}}|dk(r$t|j||dz|dzz S|dk(r?ttt j zt|j||dzzSt||Nr>r*)r+rr0rr rrrrs r7rBz erf2inv.fdiffsxyy1 q=tyy1~q(A-. . ]8AFF?3tyy1~q'8#99 9$T84 4r9c|tjus|tjurtjS|jr|jrtjS|jr"|tjurtj S|tjur|jrtjS|jr t |S|tj ur t| S|jr|S|tj ur t |S|jr'|jrtjSt |S|jr|Syr)rrKrPr.rMrLrFrR)rWr4r5s r7rZz erf2inv.eval s :aee55L YY19966M YY1:::  !%%ZAII55L YY!9  !**_A2;  YYH !**_!9  99yyvv ay 99H r9cV|j\}}|jr|jryyy)NTr )r1r4r5s r7rxzerf2inv._eval_is_zero;s&yy1 99#9r9N)rrrrrBrrZrxrr9r7rSrSs&0f54r9rSceZdZdZedZddZfdZdZdZ dZ dZ e Z e Z e Zdd Zd Zfd Zdfd Zfd ZxZS)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 cd|jrtjS|tjurtjS|tjurtjS|jrtjS|j \}}|rt |dtztz|zzSyr) rPrrNrLr.extract_branch_factorrMr r rWrurr`s r7rZzEi.evals 99%% % !**_::  !$$ $66M 99%% %'')A b6AaCF1H$ $ r9cpt|jd}|dk(rt||z St||rI)rr+rrr1rArXs r7rBzEi.fdiffs61& q=s8C< $T84 4r9c|jdtdz jr,t||t t zj |zSt||Sr8)r+rrr _eval_evalfr r )r1precrs r7rTzEi._eval_evalfsQ IIaLB ' 4 47&t,"/A/A$/GG Gw"4((r9c Vddlm}|dtd|z ttzz S)Nrrr)rrrr r rs r7rzEi._eval_rewrite_as_uppergammas)F1jnQ.//!B$66r9c Ptdtd|z ttzz S)Nr>r)rrr r rs r7rzEi._eval_rewrite_as_expints$q*R.*++ad22r9c zt|trt|jdStt |Sre)rQrlir+rrs r7_eval_rewrite_as_lizEi._eval_rewrite_as_lis. a affQi= #a&zr9c |jr%t|t|zttzz St|t|zSr) is_negativeShiChir r rs r7_eval_rewrite_as_SizEi._eval_rewrite_as_Sis6 ==q6CF?QrT) )q6CF? 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A F1 @ @r9rMcneZdZdZedZdZdZdZdZ 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|}||k7r t ||S|j r|dks|j s6d|zj r'tt ||dz z|d|z |zS|j\}}|tjury|jr^|dkDsyt ||dtztz|ztj|dz zzt|dz z t||dz zzz Stdtztz|z|zdz ||dz zz|d|z zt ||zS)Nr)gammarr*r>)rr}rrr is_IntegerrrOrr. is_integerr r rOrr)rWnurur}rnu2r`s r7rZz expint.evalcs=On 9#q> ! ==R1WR]]"?P?PjR!VZB5J)JKL L&&(1 ;  ==6"a=B$q&(1==26229R!V3DDZPQ]UWZ[U[E\\] ]!Br ! $q(!b1f+5eAFmCfRQRmS Sr9c |j\}}|dk(r!||dz z tgddgddd|z gg|zS|dk(rt|dz | St||r4)r+r'rr)r1rArrus r7rBz expint.fdiffysn A q=QK<QFQ1r6NB JJ J ]261%% %$T84 4r9c 8ddlm}||dz z|d|z |zS)Nrrr>)rr)r1rrurrs r7rz"expint._eval_rewrite_as_uppergammas#F26{:a"fa000r9c |dk(r2t|tt tzz ttzz S|jr|dkDrt | }||dz zt |dz z t|jtzt|t |dz z tt|dz Dcgc]}t ||z dz ||zzc}zzS|Scc}wrI) rMr r r r~rrE1rrrr)r1rrurr4rbs r7_eval_rewrite_as_Eizexpint._eval_rewrite_as_Eis 7qA2b5))**QrT1 1 ]]rAvAArAv;ya00Ar1BBAya((%Q-HQiQ +AqD0HIJJ JKIs8C c V|jtjtfi|Sr)rrMrrs r7rzexpint._eval_expand_funcs#'t||B''8%88r9c >|dk7r|St|t|z Sr:)r]r^)r1rrurs r7r_zexpint._eval_rewrite_as_Sis 7K1vAr9c\|jdj|s}|jd}|dk(r,|j|j}|j|||S|jr1|dkDr,|j |j}|j|||St | |||SrI)r+hasr_rrr~rr)r1r4r`rrrrtrs r7rrzexpint._eval_nseriessyy|"1BQw,D,,dii8q!T2226,D,,dii8q!T22w$Q400r9c|ddlm}|d}|jd}|tjuru|jd}t |D cgc](} tj | zt|| z|| zz *c} |d||zz |gz} t| |z t| zStt|3||||Scc} wrv) rrr+rrLrrOrrrrrr) r1r`rr4rrrrrurbrrs r7rzexpint._eval_aseriess,a YYq\ AJJ  ! AKPQR8Ta!OB$::QTATX]^_`acd`d^dfgXhWiiAGAIa( (VT0E1dCCUs -B9cddlm}|j\}}tt d|j }||| zt | |zz|dtjfSNrrdrYr>) rfrer+rrrgrrrL)r1r+rrer`r4rYs r7rjz expint._eval_rewrite_as_IntegralsW6yy1 'T277 8A2QBqD )Aq!**+=>>r9rw)rrrrrrZrBrrrr_rxryrzrrrrjrrs@r7rrs]dNTT*51 9... 1 D?r9rctd|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)rus r7rrs@ !Q<r9cveZdZdZedZddZdZdZdZ dZ dZ e Z d Z e Zd Zd Zdd ZddZdZy )rYa 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|jrtjS|tjurtjS|tj urtj S|jrtjSyr)rPrr.rMrNrLrAs r7rZzli.evalBsS 9966M !%%Z%% % !**_::  9966M r9cz|jd}|dk(rtjt|z St ||rIr+rrMrrrRs r7rBzli.fdiffM6iil q=553s8# #$T84 4r9cx|jd}|js|j|jSyre)r+r~r0rgrts r7rizli._eval_conjugateTs2 IIaL%%99Q[[]+ +&r9c 0t|tdzSr)LirYrs r7_eval_rewrite_as_Lizli._eval_rewrite_as_LiZ!ur!u}r9c *tt|Sr)rMrrs r7rzli._eval_rewrite_as_Ei]s#a&zr9c ddlm}|dt|  tjtt|ttj t|z z zztt| z SNrr)rrrrrrMrs r7rzli._eval_rewrite_as_uppergamma`s`FAAw''CF c!%%A,&7789;>Aw<H Ir9c Nttt|ztttt|zzz tj ttj t|z tt|z zz ttt|zz Sr)Cir rSirrrMrs r7r_zli._eval_rewrite_as_Siesp1SV8 qAc!fH~-AEE#a&L)CAK789;>qQx=I Jr9c tt|tt|z tjttj t|z tt|z zz Sr)r^rr]rrrMrs r7rzzli._eval_rewrite_as_ShiksJCF c#a&k)AFFCc!f 4ECPQF 4S,TTUr9c t|tddt|ztjtt|ttjt|z z zzt zS)N)r>r>)r*r*)rr&rrrMrrs r7rzli._eval_rewrite_as_hyperpsYAuVVSV44CF c!%%A,&7789;EF Gr9c tt|  tjttjt|z tt|z zz t ddt| z S)N)rr))rrr)rrrrMr'rs r7rzli._eval_rewrite_as_meijergtsYc!fW AEE#a&L(9CAK(G HH*lSVG<= >r9Nc 0|tt|zSr)rarrs r7rzli._eval_rewrite_as_tractablexs4A<r9c|jd}td|Dcgc]}t||zt||zz !}}ttt|zt |zScc}wrI)r+rrrrr)r1r4r`rrrurbrs r7rrzli._eval_nseries{s` IIaL7 2 r9rYcDeZdZdZedZd dZdZdZd dZ d dZ y) 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 c|tjurtjS|tdk(rtjSyr)rrLr.rAs r7rZzLi.evals0  ?::  !A$Y66Mr9cz|jd}|dk(rtjt|z St ||rIrrRs r7rBzLi.fdiffrr9cJ|jtj|Sr)rrYevalf)r1rUs r7rTzLi._eval_evalfs||B%%d++r9c 0t|tdz Sr)rYrs r7rZzLi._eval_rewrite_as_lirr9Nc N|jtjddS)NrT)r2)rrYrs r7rzLi._eval_rewrite_as_tractables!||B'' $'??r9cZ|j|j}|j|||Sr)rZr+rr)r1r4r`rrrts r7rrzLi._eval_nseriess+ $D $ $dii 0q!T**r9rrrw) rrrrrrZrBrTrZrrrrr9r7rrs6=@ 5,@+r9rcHeZdZdZedZddZdZdZdfd Z xZ S) TrigonometricIntegralz) Base class for trigonometric integrals. cV|tjur |jS|tjur|j S|tj ur|j S|jr |jS|jtt}|)|jddk(r|jt}||j|dS|jtt }||j|dS|jtd}|%|jddk(r|jd}||j|S|j\}}|dk(r||k(rydtztz|z|jdz||zS)Nrr>rr*)rr._atzerorL_atinfrN _atneginfrPrTrr _trigfunc_Ifactor _minusfactorrOr rPs r7rZzTrigonometricIntegral.evalsn ;;;  !**_::<  !$$ $==? " 99;;   ' ' 1 6 :#--*a/++A.B ><<A& &  ' ' A2 7 ><<B' '  ' ' 2 7 :#--*a/++B/B >##B' ''')A 6bAg tAvax a((3r722r9c|t|jd}|dk(r|j||z St||rI)rr+rrrRs r7rBzTrigonometricIntegral.fdiff s<1& q=>>#&s* *$T84 4r9c J|j|jtSr)rrrMrs r7rz)TrigonometricIntegral._eval_rewrite_as_Eis++A.66r::r9c Nddlm}|j|j|Sr)rrrrrs r7rz1TrigonometricIntegral._eval_rewrite_as_uppergammas!F++A.66zBBr9c|jdj|ddk7rt| |||S|j |j|||}|j ddk7r|dz}|j t dd}|j ddk7r|tt|zz }|j||jdj|||S)Nrr>c||z|z Srr)rYr`s r7z5TrigonometricIntegral._eval_nseries..s!Q$q&r9F) simultaneous) r+rrrrrreplacerrr)r1r4r`rr baseseriesrs r7rrz#TrigonometricIntegral._eval_nseriess 99Q<  Q "a '7(At4 4^^A&44Q4@ >>!  ! !OJ''-@u'U >>!  ! *s1v- -Jq$))A,/==aDIIr9rrw) rrrrrrZrBrrrrrrs@r7rrs6333>5;C J Jr9rceZdZdZeZejZe dZ e dZ e dZ e dZ dZdZeZdZfd 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 c0ttjzSrr rrrWs r7rz Si._atinfns!&&yr9c2t tjzSrrrs r7rz Si._atneginfrss166zr9ct| Sr)rrAs r7rzSi._minusfactorvs 1v r9c,tt|z|zSr)r r]rWrusigns r7rz Si._IfactorzsQx}r9c tdz ttt|zttt |zz dz tz zSr)r rrr rs r7rzSi._eval_rewrite_as_expint~s=!tr*Q-/*R A2q0@-AA1DQFFFr9c xddlm}ttd|gj}|t ||d|fSrc)rfrerrrgr%ris r7rjzSi._eval_rewrite_as_Integrals66 'aS166 7Q!Q++r9c<|jdj|||}|j|d}|tjur+|j |dt |jrdnd}|jr|S|js|j|S|SNrrrrr r+rrrrKrrr\rPrr0rs r7rzSi._eval_as_leading_termsiil**14d*Cxx1~ 155=99Qbh.B.Bs9LD <<J!!99T? "Kr9crddlm}|d}|tjur|jd}t |dzdzDcgc]0}tj |ztd|zz|d|zdzzz 2c}|d||zz |gz} t |dzDcgc]3}tj |ztd|zdzz|d|dzzzz 5c}|d||zz |gz} tdz t|t| zz t|t| zz Stt|;||||Scc}wcc}wr)rrrrLr+rrOrr r#rr$rrr) r1r`rr4rrrrurbpqrs r7rzSi._eval_aseriessH,a AJJ  ! A"1a4!8_.!IacN2Q1q\A.16qAvq1A0BCA#1a4[*!IacAg$66QAYG*-21QT61-=,>?Aa4#a&a.(3q6#q'>9 9R,Qq$??.*s 5D/8D4c<|jd}|jryyrlr rts r7rxzSi._eval_is_zerorr9)rrrrr$rrr.rrrrrrrrj_eval_rewrite_as_sincrrrxrrs@r7rr$sDLIffGG, 7 @ r9rceZdZdZeZejZe dZ e dZ e dZ e dZ dZdZdZfd 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 c"tjSr)rr.rs r7rz Ci._atinfs vv r9cttzSr)r r rs r7rz Ci._atneginfs t r9c4t|ttzzSrrr r rAs r7rzCi._minusfactors!uqt|r9c@t|ttzdz |zzSrr^r r rs r7rz Ci._Ifactor s1v"Qt ##r9c zttt|zttt |zz dz Sr)rrr rs r7rzCi._eval_rewrite_as_expints2JqM!O$r*aR.*:';;>r9c ddlm}ttd|gj}t j t|z|dt|z |z |d|fz Sr) rfrerrrgrrrr#ris r7rjzCi._eval_rewrite_as_IntegralsP6 'aS166 7||c!f$x3q61 q!Qi'HHHr9c|jdj|||}|j|d}|tjur+|j |dt |jrdnd}|jr;|j|\}}| t|n|}t|||zztzS|jr|j|S|Srr+rrrrKrrr\rPrmrrrsr0r1r4rrrXrrorps r7rzCi._eval_as_leading_termiil**14d*Cxx1~ 155=99Qbh.B.Bs9LD <<((+DAq!\3q6tDq6AdF?Z/ / ^^99T? "Kr9cddlm}|d}|tjtjfvr|j d}t |dzdzDcgc]0}tj|ztd|zz|d|zdzzz 2c}|d||zz |gz} t |dzDcgc]3}tj|ztd|zdzz|d|dzzzz 5c}|d||zz |gz} t|t| zt|t| zz } |tjur| ttzz } | Stt|C||||Scc}wcc}wr)rrrrLrNr+rrOrr$rr#r r rrr) r1r`rr4rrrrurbrrresultrs r7rzCi._eval_aseries&sg,a QZZ!3!34 4 ! A"1a4!8_.!IacN2Q1q\A.16qAvq1A0BCA#1a4[*!IacAg$66QAYG*-21QT61-=,>?AVS!W%AQ(88F***!B$MR,Qq$??.*s 5E+8E)rrrrr#rrrrrrrrrrrjrrrrs@r7rrsM^IG$$?I @@r9rc~eZdZdZeZejZe dZ e dZ e dZ e dZ dZdZdZy ) r]a Sinh integral. Explanation =========== This function is defined by .. math:: \operatorname{Shi}(z) = \int_0^z \frac{\sinh{t}}{t} \mathrm{d}t. It is an entire function. Examples ======== >>> from sympy import Shi >>> from sympy.abc import z The Sinh integral is a primitive of $\sinh(z)/z$: >>> Shi(z).diff(z) sinh(z)/z It is unbranched: >>> from sympy import exp_polar, I, pi >>> Shi(z*exp_polar(2*I*pi)) Shi(z) The $\sinh$ integral behaves much like ordinary $\sinh$ under multiplication by $i$: >>> Shi(I*z) I*Si(z) >>> Shi(-z) -Shi(z) It can also be expressed in terms of exponential integrals, but beware that the latter is branched: >>> from sympy import expint >>> Shi(z).rewrite(expint) expint(1, z)/2 - expint(1, z*exp_polar(I*pi))/2 - I*pi/2 See Also ======== Si: Sine integral. Ci: Cosine integral. Chi: Hyperbolic cosine integral. Ei: Exponential integral. expint: Generalised exponential integral. E1: Special case of the generalised exponential integral. li: Logarithmic integral. Li: Offset logarithmic integral. References ========== .. [1] https://en.wikipedia.org/wiki/Trigonometric_integral c"tjSrrrLrs r7rz Shi._atinf{ zzr9c"tjSr)rrNrs r7rz Shi._atneginfs!!!r9ct| Sr)r]rAs r7rzShi._minusfactors Awr9c,tt|z|zSr)r rrs r7rz Shi._IfactorsAwt|r9c t|ttttz|zz dz ttzdz z Sr)rr r r rs r7rzShi._eval_rewrite_as_expints519QrT?1,--q01R4699r9c<|jd}|jryyrlr rts r7rxzShi._eval_is_zerorr9c6|jdj|}|j|d}|tjur+|j |dt |jrdnd}|jr|S|js|j|S|S)Nrrrrrrs r7rzShi._eval_as_leading_terms~iil**1-xx1~ 155=99Qbh.B.Bs9LD <<J!!99T? "Kr9N)rrrrr"rrr.rrrrrrrrxrrr9r7r]r]8su=~IffG"": r9r]cxeZdZdZeZejZe dZ e dZ e dZ e dZ dZdZy) r^a  Cosh integral. Explanation =========== This function is defined for positive $x$ by .. math:: \operatorname{Chi}(x) = \gamma + \log{x} + \int_0^x \frac{\cosh{t} - 1}{t} \mathrm{d}t, where $\gamma$ is the Euler-Mascheroni constant. We have .. math:: \operatorname{Chi}(z) = \operatorname{Ci}\left(e^{i \pi/2}z\right) - i\frac{\pi}{2}, which holds for all polar $z$ and thus provides an analytic continuation to the Riemann surface of the logarithm. By lifting to the principal branch we obtain an analytic function on the cut complex plane. Examples ======== >>> from sympy import Chi >>> from sympy.abc import z The $\cosh$ integral is a primitive of $\cosh(z)/z$: >>> Chi(z).diff(z) cosh(z)/z It has a logarithmic branch point at the origin: >>> from sympy import exp_polar, I, pi >>> Chi(z*exp_polar(2*I*pi)) Chi(z) + 2*I*pi The $\cosh$ integral behaves somewhat like ordinary $\cosh$ under multiplication by $i$: >>> from sympy import polar_lift >>> Chi(polar_lift(I)*z) Ci(z) + I*pi/2 >>> Chi(polar_lift(-1)*z) Chi(z) + I*pi It can also be expressed in terms of exponential integrals: >>> from sympy import expint >>> Chi(z).rewrite(expint) -expint(1, z)/2 - expint(1, z*exp_polar(I*pi))/2 - I*pi/2 See Also ======== Si: Sine integral. Ci: Cosine integral. Shi: Hyperbolic sine integral. Ei: Exponential integral. expint: Generalised exponential integral. E1: Special case of the generalised exponential integral. li: Logarithmic integral. Li: Offset logarithmic integral. References ========== .. 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Explanation =========== This function is defined by .. math:: \operatorname{S}(z) = \int_0^z \sin{\frac{\pi}{2} t^2} \mathrm{d}t. It is an entire function. Examples ======== >>> from sympy import I, oo, fresnels >>> from sympy.abc import z Several special values are known: >>> fresnels(0) 0 >>> fresnels(oo) 1/2 >>> fresnels(-oo) -1/2 >>> fresnels(I*oo) -I/2 >>> fresnels(-I*oo) I/2 In general one can pull out factors of -1 and $i$ from the argument: >>> fresnels(-z) -fresnels(z) >>> fresnels(I*z) -I*fresnels(z) The Fresnel S integral obeys the mirror symmetry $\overline{S(z)} = S(\bar{z})$: >>> from sympy import conjugate >>> conjugate(fresnels(z)) fresnels(conjugate(z)) Differentiation with respect to $z$ is supported: >>> from sympy import diff >>> diff(fresnels(z), z) sin(pi*z**2/2) Defining the Fresnel functions via an integral: >>> from sympy import integrate, pi, sin, expand_func >>> integrate(sin(pi*z**2/2), z) 3*fresnels(z)*gamma(3/4)/(4*gamma(7/4)) >>> expand_func(integrate(sin(pi*z**2/2), z)) fresnels(z) We can numerically evaluate the Fresnel integral to arbitrary precision on the whole complex plane: >>> fresnels(2).evalf(30) 0.343415678363698242195300815958 >>> fresnels(-2*I).evalf(30) 0.343415678363698242195300815958*I See Also ======== fresnelc: Fresnel cosine integral. References ========== .. [1] https://en.wikipedia.org/wiki/Fresnel_integral .. [2] https://dlmf.nist.gov/7 .. [3] https://mathworld.wolfram.com/FresnelIntegrals.html .. [4] https://functions.wolfram.com/GammaBetaErf/FresnelS .. [5] The converging factors for the fresnel integrals by John W. 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A  1+61q1!IacAg$66acAg,QqS1W-AaC81Q3GI6A6AaC 1+61q1]]A- !A#'0BBacAg,QqS1W-AaC!G >> 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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