Цi3SrSSKJr SSKJr SSKJr SSKJrJ r J r SSK J r J r Jr SSKJr SSKJr SS KJr SS KJrJr SS KJr SS KJrJrJr SS KJrJ r J!r! SSK"J#r#J$r$ SSK%J&r&J'r' SSK(J)r)J*r*J+r+ SSK,J-r-J.r. SSK/J0r0J1r1J2r2 SSK3J4r4J5r5 SASjr6"SS\5r7"SS\5r8"SS\5r9"SS\5r:"SS\5r;"SS \5r<"S!S"\5r="S#S$\5r>"S%S&\5r?S'r@"S(S)\5rA"S*S+\5rB"S,S-\5rC"S.S/\C5rD"S0S1\C5rE"S2S3\C5rF"S4S5\C5rG"S6S7\5rH"S8S9\H5rI"S:S;\H5rJ"S<S=\5rK"S>S?\5rLg@)Bz|This module contains various functions that are special cases of incomplete gamma functions. It should probably be renamed. ) EulerGamma)Add)cacheit)FunctionArgumentIndexError expand_mul)IpiRational)is_eq)Pow)S)Dummyuniquely_named_symbol)sympify) factorial factorial2RisingFactorial) polar_liftre unpolarify)ceilingfloor)sqrtroot)explog exp_polar)coshsinh)cossinsinc)hypermeijergc jURSR(aAU(a(SUS'UR"U40UD6[R4$U[R4$U(a1URSR"U40UD6R 5up4OURSR 5up4UR U[U--5UR U[U-- 5-S- nUR U[U--5UR U[U-- 5- S[-- nXV4$)NrFcomplex)argsis_extended_realexpandrZero as_real_imagfuncr )selfdeephintsxyrims f/var/www/html/ai-image-ml/venv/lib/python3.13/site-packages/sympy/functions/special/error_functions.pyreal_to_real_as_real_imagr6s yy|$$ $E) KK..7 7!&&> ! yy|""4151>>@1yy|((* ))A!G tyyQqS1 11 4B ))A!G tyyQqS1 1AaC 8B 8Oc^\rSrSrSrSrSSjrSSjr\S5r \ \ S55r Sr S rS rS rS rS rSrSrSrSrSSjrSrSrSSjrU4Sjr\rSrU=r$)erf0aZ 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 TcUS:Xa/S[URSS-*5-[[5- $[ X5eNr(rrr)rr rr/argindexs r5fdiff erf.fdiffs< q=S$))A,/)**483 3$T4 4r7c[$z( Returns the inverse of this function. erfinvr?s r5inverse erf.inverses  r7cfUR(aU[RLa[R$U[RLa[R$U[R La[R $UR(a[R$[U[5(aURS$[U[5(a [RURS- $UR(a[R$[U[5(a-URSR(aURS$UR[5nU[R[R 4;aU$UR!5(a U"U*5*$gNrr=) is_NumberrNaNInfinityOneNegativeInfinity NegativeOneis_zeror, isinstancerFr)erfcinverf2invextract_multiplicativelyr could_extract_minus_signclsargts r5evalerf.evals( ==aee|uu  "uu ***}}$vv c6 " "HHQK  c7 # #55388A;& & ;;66M c7 # # (;(;88A;   ( ( + Q//0 0J  ' ' ) )I:  *r7cJUS:d US-S:Xa[R$[U5n[US- [S5- 5n[ U5S:aUS*US--US- -X-- $S[R U--X--U[ U5-[[5-- $Nrr(r=) rr,rrlenrPrrr nr2previous_termsks r5 taylor_termerf.taylor_terms q5AEQJ66M Aq1uadl#A>"Q&&r**QT1QU;QSAA))AD0!IaL.b2IJJr7cZURURSR55$Nrr.r) conjugater/s r5_eval_conjugateerf._eval_conjugate"yy1//122r7c4URSR$rhr)r*rks r5 _eval_is_realerf._eval_is_realyy|,,,r7crURSR(agURSR$NrT)r) is_finiter*rks r5_eval_is_finiteerf._eval_is_finites* 99Q< ! !99Q<00 0r7c4URSR$rhr)rQrks r5 _eval_is_zeroerf._eval_is_zeroyy|###r7c SSKJn [US-5U- [RU"[R US-5[[ 5- - -$Nr uppergammar('sympy.functions.special.gamma_functionsrrrrNHalfr r/zkwargsrs r5_eval_rewrite_as_uppergammaerf._eval_rewrite_as_uppergammas=FAqDz!|QUUZ1%=d2h%FFGGr7c [R[- U-[[5- n[R[-[ U5[[ U5-- -$NrrNr rr fresnelcfresnelsr/rrrYs r5_eval_rewrite_as_fresnelserf._eval_rewrite_as_fresnels@uuqy!mDH$ HSMAhsmO;<>>r7c [US-5U- U[[RUS-5-[[5- - $Nr(rexpintrrr rs r5_eval_rewrite_as_expinterf._eval_rewrite_as_expints6AqDz!|aqvvq!t 44T"X===r7c "SSKJn U(aWU"X[R5nU[RLa-[R [ U*5[US-*5--$[R[ U5[US-*5-- $)Nr)limitr() sympy.series.limitsrrrMrOrP_erfsrrN)r/rlimitvarrrlims r5_eval_rewrite_as_tractableerf._eval_rewrite_as_tractablesl- QZZ0Ca(((}}uaRyadU';;;uuuQxQTE ***r7c :[R[U5- $r)rrNerfcrs r5_eval_rewrite_as_erfcerf._eval_rewrite_as_erfcsuutAwr7c 6[*[[U-5-$rr erfirs r5_eval_rewrite_as_erfierf._eval_rewrite_as_erfisr$qs)|r7cDURSRXUS9nURUS5nU[RLaUR USUS:XaSOSS9nXR ;a&UR(aSU-[[5- $URU5$)Nrlogxcdirr-+dirr() r)as_leading_termsubsrComplexInfinityr free_symbolsrQrr r.r/r2rrrYarg0s r5_eval_as_leading_termerf._eval_as_leading_termsiil**1d*Cxx1~ 1$$ $99Qdbjsc9BD   T\\S5b> !99T? 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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 TcUS:Xa/S[URSS-*5-[[5- $[ X5e)Nr=r_rr(r>r?s r5rA erfc.fdiff`s< q=c499Q<?*++DH4 4$T4 4r7c[$rDrSr?s r5rG erfc.inversefs r7cUR(agU[RLa[R$U[RLa[R$UR (a[R $[U[5(a [R URS- $[U[5(aURS$UR (a[R $UR[5nU[R[R4;aU*$UR5(a SU"U*5- $g)Nrr()rKrrLrMr,rQrNrRrFr)rSrUr rOrVrWs r5r[ erfc.evalms ==aee|uu  "vv uu c6 " "55388A;& & c7 # #88A;  ;;55L  ( ( + Q//0 04K  ' ' ) )sC4y=  *r7cvUS:Xa[R$US:d US-S:Xa[R$[U5n[ US- [S5- 5n[ U5S:aUS*US--US- -X-- $S[R U--X--U[U5-[[5-- $r^) rrNr,rrr`rPrrr ras r5reerfc.taylor_terms 655L Ua!eqj66M Aq1uadl#A>"Q&&r**QT1QU;QSAA!--**QT11Yq\>$r(3JKKr7cZURURSR55$rhrirks r5rlerfc._eval_conjugaternr7c4URSR$rhrprks r5rqerfc._eval_is_realrsr7Nc JUR[5RSSUS9$N tractableT)r0rrewriter9r/rrrs r5rerfc._eval_rewrite_as_tractable#||C ((4((SSr7c :[R[U5- $r)rrNr9rs r5_eval_rewrite_as_erferfc._eval_rewrite_as_erfsuus1v~r7c V[R[[[U-5--$r)rrNr rrs r5rerfc._eval_rewrite_as_erfisuuqac{""r7c [R[- U-[[5- n[R[R[-[ U5[[ U5-- -- $rrrs r5rerfc._eval_rewrite_as_fresnelssIuuqy!mDH$uu HSMAhsmO$CDDDr7c [R[- U-[[5- n[R[R[-[ U5[[ U5-- -- $rrrs r5rerfc._eval_rewrite_as_fresnelcsIuuQwk$r("uu HSMAhsmO$CDDDr7c [RU[[5- [ [R //S/[ SS5/US-5-- $r)rrNrr r%rr rs r5rerfc._eval_rewrite_as_meijergsCuuqbz'166(Bhr1o=NPQSTPT"UUUUr7c [RSU-[[5- [ [R /S[R -/US-*5-- $r)rrNrr r$rrs r5rerfc._eval_rewrite_as_hypersAuuqs48|E166(QqvvXJA$FFFFr7c SSKJn [R[ US-5U- [RU"[R US-5[ [ 5- - -- $r)rrrrNrrr rs r5r erfc._eval_rewrite_as_uppergammasFFuutAqDz!|QUUZ1-Ed2h-N%NOOOr7c [R[US-5U- - U[[RUS-5-[[ 5- -$r)rrNrrrr rs r5rerfc._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 TcUS:Xa.S[URSS-5-[[5- $[ X5er<r>r?s r5rA erfi.fdiffs9 q=S1q))$r(2 2$T4 4r7c:UR(agU[RLa[R$UR(a[R$U[R La[R $UR(a[R$UR 5(a U"U*5*$UR[5nUbU[R La[$[U[5(a[URS-$[U[5(a'[[RURS- -$[U[5(a5URSR(a[URS-$gggrJ)rKrrLrQr,rMrVrUr rRrFr)rSrNrTrXrnzs r5r[ erfi.eval"s ;;AEEzuu vv ajjzz! 9966M % % ' 'G8O ' ' * >QZZ"f%%|#"g&&!%%"''!*,--"g&&2771:+=+=|#,>& r7c US:d US-S:Xa[R$[U5n[US- [S5- 5n[ U5S:aUSUS--US- -X-- $SX--U[ U5-[ [5-- $r^)rr,rrr`rrr ras r5reerfi.taylor_term@s q5AEQJ66M Aq1uadl#A>"Q&%b)AqD0AE:AC@@14x9Q<R!899r7cZURURSR55$rhrirks r5rlerfi._eval_conjugateMrnr7c4URSR$rhrprks r5_eval_is_extended_realerfi._eval_is_extended_realPrsr7c4URSR$rhrzrks r5r{erfi._eval_is_zeroSr}r7c JUR[5RSSUS9$rrrs r5rerfi._eval_rewrite_as_tractableVrr7c 6[*[[U-5-$r)r r9rs r5rerfi._eval_rewrite_as_erfYsr#ac({r7c B[[[U-5-[- $r)r rrs r5rerfi._eval_rewrite_as_erfc\sac{Qr7c [R[-U-[[5- n[R[- [ U5[[ U5-- -$rrrs r5rerfi._eval_rewrite_as_fresnels_rr7c [R[-U-[[5- n[R[- [ U5[[ U5-- -$rrrs r5rerfi._eval_rewrite_as_fresnelccrr7c U[[5- [[R//S/[ SS5/US-*5-$rrrs r5rerfi._eval_rewrite_as_meijerggs9bz'166(Bhr1o5FANNNr7c SU-[[5- [[R/S[R-/US-5-$rrrs r5rerfi._eval_rewrite_as_hyperjs6s48|E166(QqvvXJ1===r7c SSKJn [US-*5U- U"[RUS-*5[[ 5- [R - -$r)rrrrrr rNrs r5r erfi._eval_rewrite_as_uppergammamsAFQTE{1}j!Q$7R@155HIIr7c [US-*5U- U[[RUS-*5-[[5- - $rrrs r5rerfi._eval_rewrite_as_expintqs:QTE{1}qA!66tBx???r7c ,UR[5$rrrs r5rerfi._eval_expand_functrr7c0URSRXUS9nURUS5nXR;a&UR(aSU-[ [ 5- $UR(aURU5$URU5$)Nrrr() r)rrrrQrr rvr.rs r5rerfi._eval_as_leading_termysuiil**1d*Cxx1~   T\\S5b> ! ^^99T? "yy~r7c>SSKJn USnU[RLaURSn[ U5Vs/sH&n[ SU-S- 5SU-USU-S---- PM( snU"SXq-- U5/-n [*[US-5[[5- [U 6--$[[U];XX45$s snfr)rrrrMr)rrr rrr rrrr r/rbrr2rrrrrdrrs r5rerfi._eval_aseriess,a AJJ  ! A"1X'%AaC!G$1q1Q37|(;<%'*/!$*:);J@!-L B Br7rc|\rSrSrSrSr\S5rSrSr Sr Sr S r S r S rS rS rSrSrSrSrSrg)erf2ia 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/ cURup#US:Xa"S[US-*5-[[5- $US:Xa"S[US-*5-[[5- $[ X5e)Nr=r_r()r)rrr rr/r@r2r3s r5rA erf2.fdiffs`yy q=c1a4%j=b) ) ]S!Q$Z<R( ($T4 4r7c`[R[R[R4nU[RLdU[RLa[R$X:Xa[R$X;dX#;a[ U5[ U5- $[ U[5(a"URSU:XaURS$UR(dUUR(dDUR(aUR(d"UR(a(UR(a[ U5[ U5- $UR5nUR5nU(aU(a U"U*U*5*$U(dU(a[ U5[ U5- $grJ) rrMrOr,rLr9rRrTr)rQr* is_infiniterV)rXr2r3chksign_xsign_ys r5r[ erf2.evalszz1--qvv6 :aee55L V66M Xq6CF? " a ! !affQi1n66!9  99 Q%7%7AMM""q}}q6CF? "++-++- vQBK< q6#a&= r7cURURSR5URSR55$rJrirks r5rlerf2._eval_conjugates5yy1//1499Q<3I3I3KLLr7ctURSR=(a URSR$rJrprks r5r'erf2._eval_is_extended_reals)yy|,,N11N1NNr7c 0[U5[U5- $rr9r/r2r3rs r5rerf2._eval_rewrite_as_erfs1vAr7c 0[U5[U5- $rrrUs r5rerf2._eval_rewrite_as_erfcsAwa  r7c Z[[[U-5[[U-5- -$rrrUs r5rerf2._eval_rewrite_as_erfis"$qs)D1I%&&r7c |[U5R[5[U5R[5- $r)r9rrrUs r5rerf2._eval_rewrite_as_fresnels'1v~~h'#a&..*BBBr7c |[U5R[5[U5R[5- $r)r9rrrUs r5rerf2._eval_rewrite_as_fresnelc r^r7c |[U5R[5[U5R[5- $r)r9rr%rUs r5rerf2._eval_rewrite_as_meijerg s'1v~~g&Q)@@@r7c |[U5R[5[U5R[5- $r)r9rr$rUs r5rerf2._eval_rewrite_as_hypers'1v~~e$s1v~~e'<<tBx&GGH AJqL!%%*QVVQT":48"CC DE Fr7c |[U5R[5[U5R[5- $r)r9rrrUs r5rerf2._eval_rewrite_as_expints'1v~~f%Av(>>>r7c ,UR[5$rrrs r5rerf2._eval_expand_funcrr7c&[UR6$r)r r)rks r5r{erf2._eval_is_zerosdii  r7rN)rrrrrrArr[rlr'rrrrrrrrrrr{rrr7r5rErEsiBJ5!!0MO!'CCA=F ?!!r7rEcH\rSrSrSrS SjrS Sjr\S5rSr Sr Sr g ) rFi!a 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/ cUS:XaK[[5[URURS5S-5-[ R -$[X5eNr=rr(rr rr.r)rrrr?s r5rA erfinv.fdiffTsG q=8C $))A, 7 :;;AFFB B$T4 4r7c[$rDrTr?s r5rGerfinv.inverseZs  r7cU[RLa[R$U[RLa[R$UR(a[R $U[R La[R$[U[5(a-URSR(aURS$UR(a[R $URS5nUbE[U[5(a/URSR(aURS*$gggNrr) rrLrPrOrQr,rNrMrRr9r)r*rUrs r5r[ erfinv.evalas :55L !-- %% % YY66M !%%Z::  a  !&&)"<"<66!9  9966M ' ' + >z"c22 7T7TGGAJ; 8U2>r7c [SU- 5$Nr=rrs r5_eval_rewrite_as_erfcinverfinv._eval_rewrite_as_erfcinvwsaclr7c4URSR$rhrzrks r5r{erfinv._eval_is_zerozr}r7rNr) rrrrrrArGrr[ryr{rrr7r5rFrF!s0/d5 *$r7rFcN\rSrSrSrS SjrS Sjr\S5rSr Sr Sr S r g ) rSi~a@ 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/ cUS:XaL[[5*[URURS5S-5-[ R -$[X5erorpr?s r5rA erfcinv.fdiffsI 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/ cURup#US:Xa#[URX#5S-US-- 5$US:Xa>[[5[ R -[URX#5S-5-$[X5eNr=r()r)rr.rr rrrrGs r5rA erf2inv.fdiffspyy q=tyy~q(A-. . ]8AFF?3tyy~q'8#99 9$T4 4r7cU[RLdU[RLa[R$UR(a!UR(a[R$UR(a#U[RLa[R $U[RLa!UR(a[R$UR(a [ U5$U[R La [U*5$UR(aU$U[R La [ U5$UR(a,UR(a[R$[ U5$UR(aU$gr)rrLrQr,rNrMrFrS)rXr2r3s r5r[ erf2inv.eval s :aee55L YY19966M YY1:::  !%%ZAII55L YY!9  !**_A2;  YYH !**_!9  99yyvv ay 99H r7chURupUR(aUR(aggg)NTrz)r/r2r3s r5r{erf2inv._eval_is_zero(s$yy 99#9r7rN) rrrrrrArr[r{rrr7r5rTrTs&0f54r7rTc^\rSrSrSr\S5rSSjrSrSr Sr Sr S r \ r \ r\ rSS jrS rSU4S jjrSU4S jjrU4SjrSrU=r$)Eii1a  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 cUR(a[R$U[RLa[R$U[RLa[R$UR(a[R$UR 5up#U(a[ U5S[-[-U--$gr) rQrrOrMr,extract_branch_factorrr r rXrr rbs r5r[Ei.evals 99%% % !**_::  !$$ $66M 99%% %'') b6AaCF1H$ $ r7cp[URS5nUS:Xa[U5U- $[X5erJ)rr)rrr/r@rYs r5rAEi.fdiffs41& q=s8C< $T4 4r7cURS[S5- R(a3[R"X5[ [ -R U5-$[R"X5$ru)r)rrr _eval_evalfr r r/precs r5rEi._eval_evalfsR IIaLB ' 4 4''3qt6H6H6NN N##D//r7c VSSKJn U"S[S5U-5*[[-- $)Nrrr)rrrr r rs r5rEi._eval_rewrite_as_uppergammas)F1jnQ.//!B$66r7c P[S[S5U-5*[[-- $)Nr=r)rrr r rs r5rEi._eval_rewrite_as_expints$q*R.*++ad22r7c [U[5(a[URS5$[[ U55$rh)rRrlir)rrs r5_eval_rewrite_as_liEi._eval_rewrite_as_lis1 a  affQi= #a&zr7c UR(a%[U5[U5-[[-- $[U5[U5-$r) is_negativeShiChir r rs r5_eval_rewrite_as_SiEi._eval_rewrite_as_Sis6 ==q6CF?QrT) )q6CF? "r7c 0[U5[U5-$r)r_eisrs r5rEi._eval_rewrite_as_tractables1vQr7c SSKJn [[SU/5R5nU"[ R U-U- U[ RU45$NrIntegralrZ)sympy.integrals.integralsrrrnamerExp1rOr/rrrrZs r5_eval_rewrite_as_IntegralEi._eval_rewrite_as_IntegralsE6 'aS166 7 ! a););Q%?@@r7c>SSKJn URSRUS5nURSR XS9nUR X5nUR (aqURU5upxUc [U5OUn[U5X--[-U"U5R(a[[-- $[R- $[T U]AXUS9$)Nr)r)rr)sympyrr)rrrrQas_coeff_exponentrrrr r rr,rr) r/r2rrrx0rYcers r5rEi._eval_as_leading_terms YYq\  1 %iil**1*8wwq ::((+DA!\3q6tDq6AF?Z/4,,": :23&&: :w,Q,EEr7c>URSRUS5nUR(a+UR"UR6nUR XU5$[ TU]XU5$rh)r)rrQr _eval_nseriesrr/r2rbrrrfrs r5rEi._eval_nseriess[ YYq\  1 % ::(($))4A??1. .w$Q400r7c2>SSKJn USnU[RLa`URSn[ U5Vs/sHn[ U5Xx-- PM snU"SXq-- U5/-n [U5U- [U 6-$[[U]/XX45$s snfNrrr=) rrrrMr)rrrrrrrrBs r5rEi._eval_aseriess,a AJJ  ! A05a911&91QT61%&'AF1HQ' 'R,Qq?? :sBrrrrhr)rrrrrrr[rArrrrr_eval_rewrite_as_Ci_eval_rewrite_as_Chi_eval_rewrite_as_Shirrrrrrrrs@r5rr1srTn % %50 7 3# ... A F1 @ @r7rc|^\rSrSrSr\S5rSrSrSr Sr Sr \ r \ r \ rS U4S jjrU4S jrS rS rU=r$)ria 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 SSKJnJn [U5nX:wa [ XR5$UR (aUS::d%UR (d:SU-R (a&[[ X!S- -U"SU- U5-55$UR5up&U[RLagUR(a^US:dg[ X5S[-[-U-[RUS- --[US- 5- [U5US- --- $[S[-[-U-U-5S- X!S- --U"SU- 5-[ X5-$)Nr)gammarr(r=)rrrrr is_Integerrrrr, is_integerr r rPrr)rXnurrrnu2rbs r5r[ expint.evalPs.On 9#> ! ==R1WR]]"?P?Pj!VZB5J)JKL L&&( ;  ==6"=B$q&(1==26229R!V3DDZPQ]UWZ[U[E\\] ]!Br ! $q(!1f+5eAFmCfRmS Sr7c URup#US:Xa X2S- -*[/SS/SSSU- //U5-$US:Xa[US- U5*$[X5ero)r)r%rr)r/r@rrs r5rA expint.fdifffsh  q=QK<QFQ1r6NB JJ J ]261%% %$T4 4r7c 6SSKJn X!S- -U"SU- U5-$)Nrrr=)rr)r/rrrrs r5r"expint._eval_rewrite_as_uppergammaos!F6{:a"fa000r7c US:Xa2[U[[*[-5-5*[[-- $UR(aUS:a[ U5*nXAS- -[ US- 5- [U5R[5-[U5[ US- 5- [[US- 5Vs/sHn[ X- S- 5XE--PM sn6--$U$s snfr) rrr r rrrE1rrrr)r/rrrr2rds r5_eval_rewrite_as_Eiexpint._eval_rewrite_as_Eiss 7qA2b5))**QrT1 1 ]]rAvAAAv;ya00Ar1BBAya((%Q-H-Qi +AD0-HIJJ JKIs>C' c VUR[5R"[40UD6$r)rrrrs r5rexpint._eval_expand_funcs!||B''8%88r7c @US:waU$[U5[U5- $rx)rr)r/rrrs r5rexpint._eval_rewrite_as_Sis 7K1vAr7cn>URSRU5(dURSnUS:Xa+UR"UR6nURXU5$UR(a1US:a+UR "UR6nURXU5$[ TU] XU5$rJ)r)hasrrrrr)r/r2rbrrrrrs r5rexpint._eval_nseriessyy|""1BQw,,dii8qT2226,,dii8qT22w$Q400r7cz>SSKJn USnURSnU[RLauURSn[ U5V s/sH'n [R U -[Xy5-X-- PM) sn U"SX-- U5/-n [U*5U- [U 6-$[[U]3XX45$s sn fr) rrr)rrMrrPrrrrrr) r/rbrr2rrrrrrdrrs r5rexpint._eval_aseriess,a YYq\ AJJ  ! AKPQR8T8a!OB$::QTA8TX]^_`a`d^dfgXhWiiAGAIa( (VT01CCUs .B8cSSKJn URupE[[ SU5R 5nU"Xd*-[ U*U-5-US[R45$NrrrZr=) rrr)rrrrrrM)r/r)rrrbr2rZs r5r expint._eval_rewrite_as_IntegralsS6yy 'T277 82QBqD )Aq!**+=>>r7rr)rrrrrrr[rArrrrrrrrrrrrrs@r5rrsbdNTT*51 9... 1 D??r7rc[SU5$)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)rs r5rrs@ !Q<r7c\rSrSrSr\S5rSSjrSrSr Sr Sr S r \ r S r\rS rS rSSjrSSjrSrSrg )ria 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 cUR(a[R$U[RLa[R$U[R La[R $UR(a[R$gr)rQrr,rNrOrMrs r5r[li.eval/sS 9966M !%%Z%% % !**_::  9966M r7czURSnUS:Xa[R[U5- $[ X5erJr)rrNrrrs r5rAli.fdiff:4iil q=553s8# #$T4 4r7cURSnUR(dURUR55$grh)r)is_extended_negativer.rjr/rs r5rlli._eval_conjugateAs2 IIaL%%99Q[[]+ +&r7c 0[U5[S5-$r)Lirrs r5_eval_rewrite_as_Lili._eval_rewrite_as_LiG!ur!u}r7c *[[U55$r)rrrs r5rli._eval_rewrite_as_EiJs#a&zr7c SSKJn U"S[U5*5*[R[[U55[[R [U5- 5- --[[U5*5- $Nrr)rrrrrrNrs r5rli._eval_rewrite_as_uppergammaMs`FAAw''CF c!%%A,&7789;>Aw<H Ir7c N[[[U5-5[[[[U5-5-- [R [[R [U5- 5[[U55- -- [[[U5-5- $r)Cir rSirrrNrs r5rli._eval_rewrite_as_SiRsp1SV8 qAc!fH~-AEE#a&L)CAK789;>qQx=I Jr7c [[U55[[U55- [R[[R [U5- 5[[U55- -- $r)rrrrrrNrs r5rli._eval_rewrite_as_ShiXsJCF c#a&k)AFFCc!f 4ECPQF 4S,TTUr7c [U5[SS[U55-[R[[U55[[R[U5- 5- --[ -$)N)r=r=)r(r()rr$rrrNrrs r5rli._eval_rewrite_as_hyper]sYAuVVSV44CF c!%%A,&7789;EF Gr7c [[U5*5*[R[[R[U5- 5[[U55- -- [ SS[U5*5- $)N)rr))rrr)rrrrNr%rs r5rli._eval_rewrite_as_meijergasYc!fW AEE#a&L(9CAK(G HH*lSVG<= >r7Nc 0U[[U55-$r)rrrs r5rli._eval_rewrite_as_tractablees4A<r7cURSn[SU5Vs/sH n[U5U-[U5U-- PM" nn[[[U55-[ U6-$s snfrJ)r)rrrrr)r/r2rbrrrrdrs r5rli._eval_nserieshsa IIaL7 2 r7rcR\rSrSrSr\S5rS SjrSrSr S Sjr S S jr S r g)rira 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 cU[RLa[R$U[S5:Xa[R$gr)rrMr,rs r5r[Li.evals0  ?::  !A$Y66Mr7czURSnUS:Xa[R[U5- $[ X5erJrrs r5rALi.fdiffrr7cJUR[5RU5$r)rrevalfrs r5rLi._eval_evalfs||B%%d++r7c 0[U5[S5- $r)rrs r5rLi._eval_rewrite_as_lirr7Nc HUR[5RSSS9$)NrT)r0)rrrs r5rLi._eval_rewrite_as_tractables!||B'' $'??r7cXUR"UR6nURXU5$r)rr)r)r/r2rbrrrs r5rLi._eval_nseriess'  $ $dii 0qT**r7rrrr) rrrrrrr[rArrrrrrr7r5rrrs6=@ 5,@+r7rcV^\rSrSrSr\S5rS SjrSrSr S U4Sjjr Sr U=r $) TrigonometricIntegraliz(Base class for trigonometric integrals. ctU[RLa UR$U[RLaUR 5$U[R LaUR 5$UR(a UR$UR[[55nUc*URS5S:XaUR[5nUbURUS5$UR[[*55nUbURUS5$UR[S55nUc&URS5S:XaURS5nUbURU5$UR5up#US:XaX!:XagS[-[-U-URS5-U"U5-$)Nrr=rr()rr,_atzerorM_atinfrO _atneginfrQrUrr _trigfunc_Ifactor _minusfactorrr rs r5r[TrigonometricIntegral.evalsj ;;;  !**_::<  !$$ $==? " 99;;   ' ' 1 6 :#--*a/++A.B ><<A& &  ' ' A2 7 ><<B' '  ' ' 2 7 :#--*a/++B/B >##B' ''') 6bg tAvax a((3r722r7c|[URS5nUS:XaURU5U- $[X5erJ)rr)r(rrs r5rATrigonometricIntegral.fdiffs:1& q=>>#&s* *$T4 4r7c JURU5R[5$r)rrrrs r5r)TrigonometricIntegral._eval_rewrite_as_Eis++A.66r::r7c NSSKJn URU5RU5$r)rrrrrs r5r1TrigonometricIntegral._eval_rewrite_as_uppergammas!F++A.66zBBr7c>URSRUS5S:wa[TU] XU5$UR U5RXU5nUR S5S:waUS-nUR [ SSS9nUR S5S:waU[[U5-- nURXRS5RXU5$)Nrr=cX-U- $rr)rZrbs r55TrigonometricIntegral._eval_nseries.. s !$q&r7F) simultaneous) r)rrrr(replacer rr)r/r2rbrr baseseriesrs r5r#TrigonometricIntegral._eval_nseriess 99Q<  Q "a '7(t4 4^^A&44Q4@ >>!  ! !OJ''-@u'U >>!  ! *s1v- -Jq))A,/==aDIIr7rrr) rrrrrrr[rArrrrrrs@r5r#r#s6333>5;C J Jr7r#c^\rSrSrSr\r\Rr \ S5r \ S5r \ S5r \ S5rSrSr\rS S jrU4S jrS rS rU=r$)ria< 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 c0[[R-$rr rrrXs r5r& Si._atinf[s!&&yr7c2[*[R-$rr<r=s r5r' Si._atneginf_ss166zr7c[U5*$r)rrs r5r*Si._minusfactorcs 1v r7c,[[U5-U-$r)r rrXrsigns r5r) Si._IfactorgsQx}r7c [S- [[[5U-5[[[*5U-5- S- [- -$r)r rrr rs r5rSi._eval_rewrite_as_expintks=!tr*Q-/*R A2q0@-AA1DQFFFr7c xSSKJn [[SU/5R5nU"[ U5USU45$r)rrrrrr#rs r5rSi._eval_rewrite_as_Integralos66 'aS166 7Q!Q++r7cNURSRXUS9nURUS5nU[RLa-UR US[ U5R(aSOSS9nUR(aU$UR(dURU5$U$Nrrrrr r)rrrrLrrrrQrJr.rs r5rSi._eval_as_leading_termvsiil**1d*Cxx1~ 155=99Qbh.B.Bs9LD <<J!!99T? "Kr7cx>SSKJn USnU[RLaURSn[ US-S-5Vs/sH1n[R U-[SU-5-USU-S--- PM3 snU"SXq-- U5/-n [ US-5Vs/sH4n[R U-[SU-S-5-USUS---- PM6 snU"SXq-- U5/-n [S- [U5[U 6-- [U5[U 6-- $[[U];XX45$s snfs snfr)rrrrMr)rrPrr r!rr"rrr) r/rbrr2rrrrrdpqrs r5rSi._eval_aseriessJ,a AJJ  ! A"1a4!8_.,!IacN2Q1q\A,.16qvq1A0BCA#1a4[*(!IacAg$66QAYG(*-21QT61-=,>?Aa4#a&a.(3q6#q'>9 9R,Qq??.*s 8D2;D7cFURSnUR(aggrurzrs r5r{Si._eval_is_zerorr7rrh)rrrrrr"r(rr,r%rr&r'r*r)rr_eval_rewrite_as_sincrrr{rrrs@r5rrsDLIffGG, 7 @ r7rc^\rSrSrSr\r\Rr \ S5r \ S5r \ S5r \ S5rSrSrS S jrU4S jrS rU=r$) ria  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"[R$r)rr,r=s r5r& Ci._atinfs vv r7c[[-$r)r r r=s r5r' Ci._atneginfs t r7c4[U5[[--$rrr r rs r5r*Ci._minusfactors!uqt|r7c@[U5[[-S- U--$rrr r rDs r5r) Ci._Ifactors1v"Qt ##r7c z[[[5U-5[[[*5U-5-*S- $r)rrr rs r5rCi._eval_rewrite_as_expints2JqM!O$r*aR.*:';;>r7c SSKJn [[SU/5R5n[ R [U5-U"S[U5- U- USU45- $r) rrrrrrrrr!rs r5rCi._eval_rewrite_as_IntegralsP6 'aS166 7||c!f$x3q61 q!Qi'HHHr7cURSRXUS9nURUS5nU[RLa-UR US[ U5R(aSOSS9nUR(a:URU5upgUc [U5OUn[U5Xr--[-$UR(aURU5$U$rLr)rrrrLrrrrQrrrrvr.r/r2rrrYrrrs r5rCi._eval_as_leading_termiil**1d*Cxx1~ 155=99Qbh.B.Bs9LD <<((+DA!\3q6tDq6AF?Z/ / ^^99T? "Kr7c>SSKJn USnU[R[R4;GaUR Sn[ US-S-5Vs/sH1n[RU-[SU-5-USU-S--- PM3 snU"SXq-- U5/-n [ US-5Vs/sH4n[RU-[SU-S-5-USUS---- PM6 snU"SXq-- U5/-n [U5[U 6-[U5[U 6-- n U[RLaU [[-- n U $[[U]CXX45$s snfs snfr)rrrrMrOr)rrPrr"rr!r r rrr) r/rbrr2rrrrrdrPrQresultrs r5rCi._eval_aseriessi,a QZZ!3!34 4 ! A"1a4!8_.,!IacN2Q1q\A,.16qvq1A0BCA#1a4[*(!IacAg$66QAYG(*-21QT61-=,>?AVS!W%AQ(88F***!B$MR,Qq??.*s 8E/;E$rrh)rrrrrr!r(rrr%rr&r'r*r)rrrrrrrs@r5rrsM^IG$$?I @@r7rc\rSrSrSr\r\Rr \ S5r \ S5r \ S5r \ S5rSrSrS S jrS rg ) ri%ai 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"[R$rrrMr=s r5r& Shi._atinfh zzr7c"[R$r)rrOr=s r5r' Shi._atneginfls!!!r7c[U5*$r)rrs r5r*Shi._minusfactorps Awr7c,[[U5-U-$r)r rrDs r5r) Shi._IfactortsAwt|r7c [U5[[[[-5U-5- S- [[-S- - $r)rrr r rs r5rShi._eval_rewrite_as_expintxs519QrT?1,--q01R4699r7cFURSnUR(aggrurzrs r5r{Shi._eval_is_zero|rr7NcPURSRU5nURUS5nU[RLa-UR US[ U5R(aSOSS9nUR(aU$UR(dURU5$U$)NrrrrrMrs r5rShi._eval_as_leading_terms~iil**1-xx1~ 155=99Qbh.B.Bs9LD <<J!!99T? "Kr7rrh)rrrrrr r(rr,r%rr&r'r*r)rr{rrrr7r5rr%su=~IffG"": r7rc\rSrSrSr\r\Rr \ S5r \ S5r \ S5r \ S5rSrS S jrS rg) riaX 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 c"[R$rror=s r5r& Chi._atinfrqr7c"[R$rror=s r5r' Chi._atneginfrqr7c4[U5[[--$rr_rs r5r*Chi._minusfactors1v"}r7c@[U5[[-S- U--$rr\rDs r5r) Chi._Ifactors!uqtAvd{""r7c [*[-S- [U5[[[[-5U-5-S- - $r)r r rrrs r5rChi._eval_rewrite_as_expints7r"uQw"Q%"Yqt_Q%6"77:::r7NcURSRXUS9nURUS5nU[RLa-UR US[ U5R(aSOSS9nUR(a:URU5upgUc [U5OUn[U5Xr--[-$UR(aURU5$U$rLrfrgs r5rChi._eval_as_leading_termrir7rrh)rrrrrrr(rrr%rr&r'r*r)rrrrr7r5rrssHTIG##; r7rcP\rSrSrSrSr\S5rS SjrSr \ r Sr Sr \ rS rg ) FresnelIntegrali z%Base class for the Fresnel integrals.TcpU[RLa[R$UR(a[R$[R nUnSnUR S5nUbU*nUnSnUR [5nUbUR[-U-nUnSnU(a X "U5-$g)NFrT) rrMrrQr,rNrUr _sign)rXrprefactnewargchangedr s r5r[FresnelIntegral.eval s  ?66M 9966M%%  , ,R 0 >hGFG  , ,Q / >iik')GFG 3v;& & r7cUS:Xa9UR[R[-URSS--5$[ X5ero)r(rrr r)rr?s r5rAFresnelIntegral.fdiff( s< q=>>!&&)DIIaL!O";< <$T4 4r7c4URSR$rhrprks r5r'&FresnelIntegral._eval_is_extended_real. rsr7c4URSR$rhrzrks r5r{FresnelIntegral._eval_is_zero3 r}r7cZURURSR55$rhrirks r5rlFresnelIntegral._eval_conjugate6 rnr7rNr)rrrrrrrr[rAr'rwr{rlr6r-rrr7r5rr s>0J''<5 --O$3-Lr7rc^\rSrSrSr\r\R*r \ \ S55r Sr SrSrSrS SjrU4S jrS rU=r$) ri< a 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 crUS:a[R$[U5n[U5S:a9USn[S-*US--SU-S- -SU-SU-S--SU-S--- U-$US-US-*U--[S5SU-S- -[SU-S----SU-S-[ SU-S-5-- $) Nrr=rr(rr_rr,rr`r rrbr2rcrPs r5refresnels.taylor_term s q566M A>"Q&"2&Qq!t QqS1W-qsAaC!G}acAg/FG1LL!t1uqj(AaD2a4!8,SSKJn USnU[R[R*4;GaURSn[ SU5Vs/sHfnSU-S-U:dM[R U-[SU-S-5-SSU-S--USU-S---SSU---[SU-5-- PMh n nSSU-- /[ SU5Vs/sHlnSU-S-U:dM[R U-[SU-S- 5-SSU-S--USU-S---SSU-S- --[SU-S- 5-- PMn sn-n U V s/sHn [S[- 5*U -PM n n U V s/sHn [S[- 5*U -PM n n U[RLaSOSn U [R-[US-5[U 6-[US-5[U 6--RU[S[- 5U-5-U"SXq-- U5-$[T U]AXX45$s snfs snfs sn fs sn fNrrrrr=r(r)rrrrMr)rrPrrr rr"rr!rrrr/rbrr2rrrrrdrPrQrZrrs r5rfresnels._eval_aseries s`,a QZZ!**- - ! A  1+6%Q1q1I!IacAg$66acAg,QqS1W-AaC81Q3GI% 6AaC 1+6%Q1q1QQ]]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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