ó ¡¼™\c@ sÿdZddlmZmZddlmZmZmZmZm Z m Z ddl m Z ddl mZmZddlmZddlmZddlmZdd lmZmZdd lmZmZdd lmZdd lmZm Z dd l!m"Z"m#Z#m$Z$ddl%m&Z&m'Z'defd„ƒYZ(defd„ƒYZ)defd„ƒYZ*defd„ƒYZ+defd„ƒYZ,defd„ƒYZ-defd„ƒYZ.defd„ƒYZ/defd „ƒYZ0d!„Z1d"efd#„ƒYZ2d$efd%„ƒYZ3d&efd'„ƒYZ4d(e4fd)„ƒYZ5d*e4fd+„ƒYZ6d,e4fd-„ƒYZ7d.e4fd/„ƒYZ8d0efd1„ƒYZ9d2e9fd3„ƒYZ:d4e9fd5„ƒYZ;d6efd7„ƒYZ<d8efd9„ƒYZ=d:S(;s This module contains various functions that are special cases of incomplete gamma functions. It should probably be renamed. iÿÿÿÿ(tprint_functiontdivision(tAddtStsympifytcacheittpitI(trange(tFunctiontArgumentIndexError(tSymbol(t factorial(tfloor(tsqrttroot(texptlog(t polar_lift(tcoshtsinh(tcostsintsinc(thypertmeijergterfcB sÈeZdZeZdd„Zdd„Zed„ƒZe e d„ƒƒZ d„Z d„Z d„Zd „Zd „Zd „Zd „Zd „Zd„Zd„Zd„Zd„Zed„ZRS(s The Gauss error function. This function is defined as: .. math :: \mathrm{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} \mathrm{d}t. Examples ======== >>> from sympy import I, oo, erf >>> from sympy.abc import z Several special values are known: >>> erf(0) 0 >>> erf(oo) 1 >>> erf(-oo) -1 >>> erf(I*oo) oo*I >>> erf(-I*oo) -oo*I In general one can pull out factors of -1 and I from the argument: >>> erf(-z) -erf(z) The error function obeys the mirror symmetry: >>> from sympy import conjugate >>> conjugate(erf(z)) erf(conjugate(z)) Differentiation with respect to z is supported: >>> from sympy import diff >>> diff(erf(z), z) 2*exp(-z**2)/sqrt(pi) We can numerically evaluate the error function to arbitrary precision on the whole complex plane: >>> erf(4).evalf(30) 0.999999984582742099719981147840 >>> erf(-4*I).evalf(30) -1296959.73071763923152794095062*I See Also ======== erfc: Complementary error function. erfi: Imaginary error function. erf2: Two-argument error function. erfinv: Inverse error function. erfcinv: Inverse Complementary error function. erf2inv: Inverse two-argument error function. References ========== .. [1] https://en.wikipedia.org/wiki/Error_function .. [2] http://dlmf.nist.gov/7 .. [3] http://mathworld.wolfram.com/Erf.html .. [4] http://functions.wolfram.com/GammaBetaErf/Erf icC sF|dkr3dt|jdd ƒttjƒSt||ƒ‚dS(Niii(RtargsRRtPiR (tselftargindex((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pytfdiffds 'cC stS(s7 Returns the inverse of this function. 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The function is defined as: .. math :: \mathrm{erfc}(x) = \frac{2}{\sqrt{\pi}} \int_x^\infty e^{-t^2} \mathrm{d}t Examples ======== >>> from sympy import I, oo, erfc >>> from sympy.abc import z Several special values are known: >>> erfc(0) 1 >>> erfc(oo) 0 >>> erfc(-oo) 2 >>> erfc(I*oo) -oo*I >>> erfc(-I*oo) oo*I The error function obeys the mirror symmetry: >>> from sympy import conjugate >>> conjugate(erfc(z)) erfc(conjugate(z)) Differentiation with respect to z is supported: >>> from sympy import diff >>> diff(erfc(z), z) -2*exp(-z**2)/sqrt(pi) It also follows >>> erfc(-z) 2 - erfc(z) We can numerically evaluate the complementary error function to arbitrary precision on the whole complex plane: >>> erfc(4).evalf(30) 0.0000000154172579002800188521596734869 >>> erfc(4*I).evalf(30) 1.0 - 1296959.73071763923152794095062*I See Also ======== erf: Gaussian error function. erfi: Imaginary error function. erf2: Two-argument error function. erfinv: Inverse error function. erfcinv: Inverse Complementary error function. erf2inv: Inverse two-argument error function. References ========== .. [1] https://en.wikipedia.org/wiki/Error_function .. [2] http://dlmf.nist.gov/7 .. [3] http://mathworld.wolfram.com/Erfc.html .. [4] http://functions.wolfram.com/GammaBetaErf/Erfc icC sF|dkr3dt|jdd ƒttjƒSt||ƒ‚dS(Niiþÿÿÿii(RRRRRR (RR((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR%s 'cC stS(s7 Returns the inverse of this function. 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The function erfi is defined as: .. math :: \mathrm{erfi}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{t^2} \mathrm{d}t Examples ======== >>> from sympy import I, oo, erfi >>> from sympy.abc import z Several special values are known: >>> erfi(0) 0 >>> erfi(oo) oo >>> erfi(-oo) -oo >>> erfi(I*oo) I >>> erfi(-I*oo) -I In general one can pull out factors of -1 and I from the argument: >>> erfi(-z) -erfi(z) >>> from sympy import conjugate >>> conjugate(erfi(z)) erfi(conjugate(z)) Differentiation with respect to z is supported: >>> from sympy import diff >>> diff(erfi(z), z) 2*exp(z**2)/sqrt(pi) We can numerically evaluate the imaginary error function to arbitrary precision on the whole complex plane: >>> erfi(2).evalf(30) 18.5648024145755525987042919132 >>> erfi(-2*I).evalf(30) -0.995322265018952734162069256367*I See Also ======== erf: Gaussian error function. erfc: Complementary error function. erf2: Two-argument error function. erfinv: Inverse error function. erfcinv: Inverse Complementary error function. erf2inv: Inverse two-argument error function. References ========== .. [1] https://en.wikipedia.org/wiki/Error_function .. [2] http://mathworld.wolfram.com/Erfi.html .. [3] http://functions.wolfram.com/GammaBetaErf/Erfi icC sE|dkr2dt|jddƒttjƒSt||ƒ‚dS(Niii(RRRRRR (RR((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyRás &cC s|jrN|tjkrtjS|tjkr5tjS|tjkrNtjSn|jƒrf|| ƒ S|jtƒ}|dk r|tjkr”tSt |t ƒr²t|j dSt |t ƒr×ttj |j dSt |tƒr|j dtjkrt|j dSndS(Nii(R"RR#R(R$R.R,RtNoneR)R RR*R%R+(R/RAtnz((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR2çs&     %cG s¦|dks|ddkr#tjSt|ƒ}t|dtdƒƒ}t|ƒdkr{|d|d|d||Sd|||t|ƒttjƒSdS(Niiiiþÿÿÿ(RR(RR R3R RR(R4R5R6R7((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR8s  cC s|j|jdjƒƒS(Ni(R9RR:(R((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR;scC s|jdjS(Ni(RR<(R((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR=scK s|jtƒjddtƒS(NRhR[(RiRRd(RRARB((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyRMscK st tt|ƒS(N(RR(RRARB((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyRjscK sttt|ƒtS(N(RRN(RRARB((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyROscK sDtjtj|ttƒ}tjtjt|ƒtt|ƒS(N(RR%R-RRRDRRE(RRARBR0((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyRFscK sDtjtj|ttƒ}tjtjt|ƒtt|ƒS(N(RR%R-RRRDRRE(RRARBR0((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyRG!scK s9|ttƒttjggdgtj g|d ƒS(Nii(RRRRR@(RRARB((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyRH%scK s6d|ttƒttjgdtjg|dƒS(Nii(RRRRR@(RRARB((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyRI(scK sLddlm}t|d ƒ||tj|d ƒttjƒtjS(Niÿÿÿÿ(R>i(R?R>RRR@RR%(RRARBR>((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyRC+scK s9t|d ƒ||ttj|d ƒttjƒS(Ni(RRJRR@R(RRARB((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyRK/scK s |jtƒS(N(RiR(RR\((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyRk2scK s:|jdjrI|r9t|d<|j||tjfS|tjfSn|rw|jdj||jƒ\}}n|jdjƒ\}}|d |d}tj|j||t |ƒƒ|j||t |ƒƒ}|d|t |ƒ|j||t |ƒƒ|j||t |ƒƒ}||fS(NiRWi( RR<RXRYRR(RZR@R9R(RR[R\R5R]R^R_R`((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyRZ5s (?/(RaRbRcRdReRRfR2RgRR8R;R=RMRjRORFRGRHRIRCRKRkRZ(((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyRP›s&B             terf2cB s’eZdZd„Zed„ƒZd„Zd„Zd„Zd„Z d„Z d„Z d „Z d „Z d „Zd „Zd „Zd„ZRS(s Two-argument error function. 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 I, oo, erf2 >>> from sympy.abc import x, y Several special values are known: >>> erf2(0, 0) 0 >>> erf2(x, x) 0 >>> erf2(x, oo) 1 - erf(x) >>> erf2(x, -oo) -erf(x) - 1 >>> erf2(oo, y) erf(y) - 1 >>> erf2(-oo, y) erf(y) + 1 In general one can pull out factors of -1: >>> erf2(-x, -y) -erf2(x, y) The error function obeys the mirror symmetry: >>> from sympy import conjugate >>> conjugate(erf2(x, y)) erf2(conjugate(x), conjugate(y)) Differentiation with respect to x, y is supported: >>> from sympy import diff >>> diff(erf2(x, y), x) -2*exp(-x**2)/sqrt(pi) >>> diff(erf2(x, y), y) 2*exp(-y**2)/sqrt(pi) See Also ======== erf: Gaussian error function. erfc: Complementary error function. erfi: Imaginary error function. erfinv: Inverse error function. erfcinv: Inverse Complementary error function. erf2inv: Inverse two-argument error function. References ========== .. [1] http://functions.wolfram.com/GammaBetaErf/Erf2/ cC sz|j\}}|dkr;dt|d ƒttjƒS|dkrgdt|d ƒttjƒSt||ƒ‚dS(Niiþÿÿÿi(RRRRRR (RRR5R]((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR‡s     cC s4tj}tj}tj}|tjks9|tjkr@tjS||krStjS||ks›||ks›||ks›||ks›||ks›||kr¯t|ƒt|ƒSt|tƒrÜ|jd|krÜ|jdS|j ƒ}|j ƒ}|r|r|| | ƒ S|s|r0t|ƒt|ƒSdS(Nii( RR$R&R(R#RR)R+RR.(R/R5R]RtNtOtsign_xtsign_y((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR2s"    H"     cC s*|j|jdjƒ|jdjƒƒS(Nii(R9RR:(R((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR;§scC s|jdjo|jdjS(Nii(RR<(R((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR=ªscK st|ƒt|ƒS(N(R(RR5R]RB((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyRj­scK st|ƒt|ƒS(N(RN(RR5R]RB((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyRO°scK s ttt|ƒtt|ƒS(N(RRP(RR5R]RB((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyRQ³scK s&t|ƒjtƒt|ƒjtƒS(N(RRiRE(RR5R]RB((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyRF¶scK s&t|ƒjtƒt|ƒjtƒS(N(RRiRD(RR5R]RB((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyRG¹scK s&t|ƒjtƒt|ƒjtƒS(N(RRiR(RR5R]RB((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyRH¼scK s&t|ƒjtƒt|ƒjtƒS(N(RRiR(RR5R]RB((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyRI¿scK s„ddlm}t|dƒ|tj|tj|dƒttjƒt|dƒ|tj|tj|dƒttjƒS(Niÿÿÿÿ(R>i(R?R>RRR%R@R(RR5R]RBR>((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyRCÂs9cK s&t|ƒjtƒt|ƒjtƒS(N(RRiRJ(RR5R]RB((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyRKÇscK s |jtƒS(N(RiR(RR\((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyRkÊs(RaRbRcRRfR2R;R=RjRORQRFRGRHRIRCRKRk(((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyRnGs=            R cB s>eZdZdd„Zdd„Zed„ƒZd„ZRS(sU 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 I, oo, erfinv >>> from sympy.abc import x Several special values are known: >>> erfinv(0) 0 >>> erfinv(1) oo Differentiation with respect to x is supported: >>> from sympy import diff >>> diff(erfinv(x), x) sqrt(pi)*exp(erfinv(x)**2)/2 We can numerically evaluate the inverse error function to arbitrary precision on [-1, 1]: >>> erfinv(0.2).evalf(30) 0.179143454621291692285822705344 See Also ======== erf: Gaussian error function. erfc: Complementary error function. erfi: Imaginary error function. erf2: Two-argument error function. erfcinv: Inverse Complementary error function. erf2inv: Inverse two-argument error function. References ========== .. [1] https://en.wikipedia.org/wiki/Error_function#Inverse_functions .. [2] http://functions.wolfram.com/GammaBetaErf/InverseErf/ icC sQ|dkr>ttjƒt|j|jdƒdƒtjSt||ƒ‚dS(Niii(RRRRR9RR@R (RR((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyRs 2cC stS(s7 Returns the inverse of this function. (R(RR((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR!scC sÌ|tjkrtjS|tjkr,tjS|tjkrBtjS|tjkrXtjSt|tƒr‚|j dj r‚|j dS|j dƒ}|dk rÈt|tƒrÈ|j dj rÈ|j d SdS(Niiÿÿÿÿ( RR#R'R&R(R%R$R)RRR<R,Rl(R/RARm((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR2 s +cK std|ƒS(Ni(R*(RRARB((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyt_eval_rewrite_as_erfcinvs(RaRbRcRR!RfR2Rs(((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR Îs /  R*cB s>eZdZdd„Zdd„Zed„ƒZd„ZRS(s· Inverse Complementary Error Function. The erfcinv function is defined as: .. math :: \mathrm{erfc}(x) = y \quad \Rightarrow \quad \mathrm{erfcinv}(y) = x Examples ======== >>> from sympy import I, oo, erfcinv >>> from sympy.abc import x Several special values are known: >>> erfcinv(1) 0 >>> erfcinv(0) oo Differentiation with respect to x is supported: >>> from sympy import diff >>> diff(erfcinv(x), x) -sqrt(pi)*exp(erfcinv(x)**2)/2 See Also ======== erf: Gaussian error function. erfc: Complementary error function. erfi: Imaginary error function. erf2: Two-argument error function. erfinv: Inverse error function. erf2inv: Inverse two-argument error function. References ========== .. [1] https://en.wikipedia.org/wiki/Error_function#Inverse_functions .. [2] http://functions.wolfram.com/GammaBetaErf/InverseErfc/ icC sR|dkr?ttjƒ t|j|jdƒdƒtjSt||ƒ‚dS(Niii(RRRRR9RR@R (RR((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyROs 3cC stS(s7 Returns the inverse of this function. (RN(RR((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR!UscC sY|tjkrtjS|tjkr,tjS|tjkrBtjS|dkrUtjSdS(Ni(RR#R(R$R%R&(R/RA((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR2[s cK std|ƒS(Ni(R (RRARB((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyt_eval_rewrite_as_erfinvfs(RaRbRcRR!RfR2Rt(((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR*#s )   R+cB s&eZdZd„Zed„ƒZRS(s@ 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 I, oo, erf2inv, erfinv, erfcinv >>> from sympy.abc import x, y Several special values are known: >>> erf2inv(0, 0) 0 >>> erf2inv(1, 0) 1 >>> erf2inv(0, 1) oo >>> erf2inv(0, y) erfinv(y) >>> erf2inv(oo, y) erfcinv(-y) Differentiation with respect to x and y is supported: >>> from sympy import diff >>> diff(erf2inv(x, y), x) exp(-x**2 + erf2inv(x, y)**2) >>> diff(erf2inv(x, y), y) sqrt(pi)*exp(erf2inv(x, y)**2)/2 See Also ======== erf: Gaussian error function. erfc: Complementary error function. erfi: Imaginary error function. erf2: Two-argument error function. erfinv: Inverse error function. erfcinv: Inverse complementary error function. References ========== .. [1] http://functions.wolfram.com/GammaBetaErf/InverseErf2/ cC sŠ|j\}}|dkr=t|j||ƒd|dƒS|dkrwttjƒtjt|j||ƒdƒSt||ƒ‚dS(Nii(RRR9RRRR@R (RRR5R]((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyRs  " .cC s÷|tjks|tjkr%tjS|tjkrJ|tjkrJtjS|tjkro|tjkrotjS|tjkr”|tjkr”tjS|tjkr­t|ƒS|tjkrÇt| ƒS|tjkrÚ|S|tjkrót|ƒSdS(N(RR#R(R%R$R R*(R/R5R]((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR2¦s   (RaRbRcRRfR2(((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR+js0 tEicB szeZdZed„ƒZdd„Zd„Zd„Zd„Zd„Z d„Z e Z e Z e Z d „Zd „ZRS( sí The classical exponential integral. 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 uppergamma, 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. sympy.functions.special.gamma_functions.uppergamma: Upper incomplete gamma function. References ========== .. [1] http://dlmf.nist.gov/6.6 .. [2] https://en.wikipedia.org/wiki/Exponential_integral .. [3] Abramowitz & Stegun, section 5: http://people.math.sfu.ca/~cbm/aands/page_228.htm cC sx|tjkrtjS|tjkr,tjS|tjkrBtjS|jƒ\}}|rtt|ƒdtt|SdS(Ni(RR(R&R$textract_branch_factorRuRR(R/RARmR4((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR2sicC sPddlm}||jdƒ}|dkr=t|ƒ|St||ƒ‚dS(Niÿÿÿÿ(t unpolarifyii(R?RwRRR (RRRwR0((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR s  cC sK|jdtdƒjr;tj||ƒttj|ƒStj||ƒS(Niiÿÿÿÿ(RRt is_positiveR t _eval_evalfRR(Rtprec((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyRy(s!cK s0ddlm}|dtdƒ|ƒ ttS(Niÿÿÿÿ(R>i(R?R>RRR(RRARBR>((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyRC-scK s tdtdƒ|ƒ ttS(Niiÿÿÿÿ(RJRRR(RRARB((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyRK3scK s0t|tƒr t|jdƒStt|ƒƒS(Ni(R)RtliRR(RRARB((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyt_eval_rewrite_as_li6scK st|ƒt|ƒS(N(tShitChi(RRARB((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyt_eval_rewrite_as_Si?scK st|ƒt|ƒS(N(Rt_eis(RRARB((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyRMEscC si|jdj|dƒ}|tjkrM|j|jŒ}|j|||ƒStt|ƒj|||ƒS(Ni(RtlimitRR(Rt _eval_nseriestsuperRu(RR5R4tlogxtx0tf((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR‚Hs (RaRbRcRfR2RRyRCRKR|Rt_eval_rewrite_as_Cit_eval_rewrite_as_Chit_eval_rewrite_as_ShiRMR‚(((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyRu¾sR      RJcB sneZdZed„ƒZd„Zd„Zd„Zd„Zd„Z e Z e Z e Z d„Z d„ZRS( s² Generalized exponential integral. 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 :math:`z` with positive real part we have .. math:: \operatorname{E}_\nu(z) = \int_1^\infty \frac{e^{-zt}}{z^\nu} \mathrm{d}t, which explains the name. The representation as an incomplete gamma function provides an analytic continuation for :math:`\operatorname{E}_\nu(z)`. If :math:`\nu` is a non-positive integer the exponential integral is thus an unbranched function of :math:`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 explains further 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. sympy.functions.special.gamma_functions.uppergamma References ========== .. [1] http://dlmf.nist.gov/8.19 .. [2] http://functions.wolfram.com/GammaBetaErf/ExpIntegralE/ .. [3] https://en.wikipedia.org/wiki/Exponential_integral c C suddlm}m}m}m}m}m}||ƒ} || krSt| |ƒS|jrh|dks|j r¨d|jr¨||||d|d||ƒƒƒS|j ƒ\}} | dkrÊdS|j r,|dksãdSt||ƒdt t | d|d||dƒ||ƒ|dS|dt t || ƒd||d|d|ƒt||ƒSdS(Niÿÿÿÿ(Rwt expand_mulR>RtgammaR iii( R?RwRŠR>RR‹R RJt is_IntegerRvt is_integerRR( R/tnuRARwRŠR>RR‹R tnu2R4((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR2µs.   ,)    =cC s’ddlm}|j\}}|dkra||d |gddgddd|gg|ƒS|dkrt|d|ƒ St||ƒ‚dS(Niÿÿÿÿ(Riii(R?RRRJR (RRRRŽRA((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyRÌs 6 cK s-ddlm}||d|d||ƒS(Niÿÿÿÿ(R>i(R?R>(RRŽRARBR>((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyRCÖsc K süddlm}m}m}m}|dkrPt||t tƒƒ ttS|jrô|dkrô||ƒ }||d||dƒt |ƒj tƒ||ƒ||dƒt gt |dƒD]"} ||| dƒ|| ^qÉŒS|SdS(Niÿÿÿÿ(t exp_polarRwRR ii( R?RRwRR RuRRRŒtE1RiRR( RRŽRARBRRwRR R5R7((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyt_eval_rewrite_as_EiÚs" " ,?cK s|jtƒjt|S(N(RiRuRJ(RR\((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyRkçscK s$|dkr|St|ƒt|ƒS(Ni(R}R~(RRŽRARB((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyRês cC s­|jdj|ƒs‘|jd}|dkrT|j|jŒ}|j|||ƒS|jr‘|dkr‘|j|jŒ}|j|||ƒSntt|ƒj|||ƒS(Nii(RthasRR‚RŒR’RƒRJ(RR5R4R„RŽR†((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR‚òs  cC s9ddlj}|j|jdjƒ|jdjƒƒS(Niÿÿÿÿii(tsage.alltalltexp_integral_eRt_sage_(Rtsage((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR—ýs(RaRbRcRfR2RRCR’RkRR‡RˆR‰R‚R—(((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyRJPsb    cC s td|ƒS(s‘ Classical case of the generalized exponential integral. This is equivalent to ``expint(1, z)``. 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. i(RJ(RA((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR‘sR{cB s†eZdZed„ƒZdd„Zd„Zd„Zd„Zd„Z d„Z e Z d „Z e Z d „Zd „Zd „ZRS( s The classical logarithmic integral. For the 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: The logarithmic integral can also be defined in terms of Ei: >>> from sympy import Ei >>> li(z).rewrite(Ei) Ei(log(z)) >>> diff(li(z).rewrite(Ei), z) 1/log(z) We can numerically evaluate the logarithmic integral to arbitrary precision on the whole complex plane (except the singular points): >>> li(2).evalf(30) 1.04516378011749278484458888919 >>> li(2*I).evalf(30) 1.0652795784357498247001125598 + 3.08346052231061726610939702133*I We can even compute Soldner's constant by the help of mpmath: >>> from mpmath import findroot >>> findroot(li, 2) 1.45136923488338 Further transformations include rewriting `li` in terms of the trigonometric integrals `Si`, `Ci`, `Shi` and `Chi`: >>> from sympy import Si, Ci, Shi, Chi >>> li(z).rewrite(Si) -log(I*log(z)) - log(1/log(z))/2 + log(log(z))/2 + Ci(I*log(z)) + Shi(log(z)) >>> li(z).rewrite(Ci) -log(I*log(z)) - log(1/log(z))/2 + log(log(z))/2 + Ci(I*log(z)) + Shi(log(z)) >>> li(z).rewrite(Shi) -log(1/log(z))/2 + log(log(z))/2 + Chi(log(z)) - Shi(log(z)) >>> li(z).rewrite(Chi) -log(1/log(z))/2 + log(log(z))/2 + Chi(log(z)) - Shi(log(z)) See Also ======== Li: Offset logarithmic integral. Ei: Exponential integral. expint: Generalised exponential integral. E1: Special case of the generalised exponential integral. Si: Sine integral. Ci: Cosine integral. Shi: Hyperbolic sine integral. Chi: Hyperbolic cosine integral. References ========== .. [1] https://en.wikipedia.org/wiki/Logarithmic_integral .. [2] http://mathworld.wolfram.com/LogarithmicIntegral.html .. [3] http://dlmf.nist.gov/6 .. [4] http://mathworld.wolfram.com/SoldnersConstant.html cC sF|tjkrtjS|tjkr,tjS|tjkrBtjSdS(N(RR(R%R&R$(R/RA((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR2qs icC s=|jd}|dkr*tjt|ƒSt||ƒ‚dS(Nii(RRR%RR (RRR0((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyRzs  cC s6|jd}|jo|js2|j|jƒƒSdS(Ni(RR<t is_negativeR9R:(RRA((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR;s cK st|ƒtdƒS(Ni(tLiR{(RRARB((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyt_eval_rewrite_as_Li‡scK stt|ƒƒS(N(RuR(RRARB((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR’ŠscK sdddlm}|dt|ƒ ƒ tjtt|ƒƒttjt|ƒƒtt|ƒ ƒS(Niÿÿÿÿ(R>i(R?R>RRR@R%(RRARBR>((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyRCscK snttt|ƒƒtttt|ƒƒtjttjt|ƒƒtt|ƒƒttt|ƒƒS(N(tCiRRtSiRR@R%(RRARB((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR’scK sNtt|ƒƒtt|ƒƒtjttjt|ƒƒtt|ƒƒS(N(R~RR}RR@R%(RRARB((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR‰˜scK sUt|ƒtddt|ƒƒtjtt|ƒƒttjt|ƒƒtjS(Nii(ii(ii(RRRR@R%t EulerGamma(RRARB((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyRIscK sctt|ƒ ƒ tjttjt|ƒƒtt|ƒƒtddfddft|ƒ ƒS(Nii((i(ii((RRR@R%R(RRARB((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyRH¡scK s|tt|ƒƒS(N(R€R(RRARB((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyRM¥s(RaRbRcRfR2RR;R›R’RCRR‡R‰RˆRIRHRM(((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR{sW         RšcB sDeZdZed„ƒZdd„Zd„Zd„Zd„ZRS(sG The offset logarithmic integral. For the use in SymPy, this function is defined as .. math:: \operatorname{Li}(x) = \operatorname{li}(x) - \operatorname{li}(2) Examples ======== >>> from sympy import I, oo, Li >>> from sympy.abc import z The following special value is known: >>> Li(2) 0 Differentiation with respect to z is supported: >>> from sympy import diff >>> diff(Li(z), z) 1/log(z) The shifted logarithmic integral can be written in terms of `li(z)`: >>> from sympy import li >>> Li(z).rewrite(li) li(z) - li(2) We can numerically evaluate the logarithmic integral to arbitrary precision on the whole complex plane (except the singular points): >>> Li(2).evalf(30) 0 >>> Li(4).evalf(30) 1.92242131492155809316615998938 See Also ======== li: Logarithmic integral. Ei: Exponential integral. expint: Generalised exponential integral. E1: Special case of the generalised exponential integral. Si: Sine integral. Ci: Cosine integral. Shi: Hyperbolic sine integral. Chi: Hyperbolic cosine integral. References ========== .. [1] https://en.wikipedia.org/wiki/Logarithmic_integral .. [2] http://mathworld.wolfram.com/LogarithmicIntegral.html .. [3] http://dlmf.nist.gov/6 cC s4|tjkrtjS|dtjkr0tjSdS(Ni(RR$R%R((R/RA((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR2æsicC s=|jd}|dkr*tjt|ƒSt||ƒ‚dS(Nii(RRR%RR (RRR0((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyRís  cC s|jtƒj|ƒS(N(RiR{tevalf(RRz((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyRyôscK st|ƒtdƒS(Ni(R{(RRARB((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR|÷scK s|jtƒjddtƒS(NRhR[(RiR{Rd(RRARB((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyRMús( RaRbRcRfR2RRyR|RM(((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyRš©s :   tTrigonometricIntegralcB sDeZdZed„ƒZdd„Zd„Zd„Zd„ZRS(s) Base class for trigonometric integrals. cC s‘|dkr|jS|tjkr,|jƒS|tjkrE|jƒS|jttƒƒ}|dkr|j dƒdkr|jtƒ}n|dk r©|j |dƒS|jtt ƒƒ}|dk rÛ|j |dƒS|jtdƒƒ}|dkr#|j dƒdkr#|jdƒ}n|dk r<|j |ƒS|j ƒ\}}|dkrj||krjdSdtt||j dƒ||ƒS(Niiiÿÿÿÿi(t_atzeroRR$t_atinfR&t _atneginfR,RRRlt _trigfunct_Ifactort _minusfactorRvR(R/RARmR4((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR2s.   !  !  icC sDddlm}||jdƒ}|dkr@|j|ƒ|SdS(Niÿÿÿÿ(Rwii(R?RwRR¤(RRRwR0((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR"s cK s|j|ƒjtƒS(N(RKRiRu(RRARB((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR’(scK s&ddlm}|j|ƒj|ƒS(Niÿÿÿÿ(R>(R?R>RKRi(RRARBR>((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyRC+scC sddlm}m}m}|d7}|jdj|dƒdkratt|ƒj|||ƒS|j |ƒj|||ƒ}|j dƒdkr¡|d8}n|j |d„dt ƒ}|j dƒdkrè||||ƒ7}n|j||jdƒj|||ƒS(Niÿÿÿÿ(RRžtPowiicS s |||S(N((R1R4((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyt8tt simultaneous( R?RRžR§RtsubsRƒR R‚R¤treplaceRX(RR5R4R„RRžR§t baseseries((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR‚/s  ( RaRbRcRfR2RR’RCR‚(((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR s    RcB sweZdZeZedƒZed„ƒZed„ƒZ ed„ƒZ ed„ƒZ d„Z d„Z d„ZRS( sÒ Sine integral. 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 icC s ttjS(N(RRR@(R/((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR¢…scC s t tjS(N(RRR@(R/((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR£‰scC s t|ƒ S(N(R(R/RA((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR¦scC stt|ƒ|S(N(RR}(R/RAtsign((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR¥‘scK s9tdtttƒ|ƒttt ƒ|ƒdtS(Ni(RR‘RR(RRARB((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyRK•scK s>ddlm}tddtƒ}|t|ƒ|d|fƒS(Niÿÿÿÿ(tIntegralR1tDummyi(R?R¯R RdR(RRARBR¯R1((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyt_eval_rewrite_as_sinc™scC s)ddlj}|j|jdjƒƒS(Niÿÿÿÿi(R”R•t sin_integralRR—(RR˜((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR—žs(RaRbRcRR¤RR¡RfR¢R£R¦R¥RKR±R—(((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR>sB   RœcB skeZdZeZejZed„ƒZ ed„ƒZ ed„ƒZ ed„ƒZ d„Z d„ZRS(s¼ Cosine integral. 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 cC stjS(N(RR((R/((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR¢ñscC sttS(N(RR(R/((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR£õscC st|ƒttS(N(RœRR(R/RA((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR¦ùscC st|ƒttd|S(Ni(R~RR(R/RAR®((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR¥ýscK s.tttƒ|ƒttt ƒ|ƒ dS(Ni(R‘RR(RRARB((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyRKscC s)ddlj}|j|jdjƒƒS(Niÿÿÿÿi(R”R•t cos_integralRR—(RR˜((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR—s(RaRbRcRR¤RtComplexInfinityR¡RfR¢R£R¦R¥RKR—(((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyRœ¢sJ  R}cB sneZdZeZedƒZed„ƒZed„ƒZ ed„ƒZ ed„ƒZ d„Z d„Z RS(sì Sinh integral. 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 icC stjS(N(RR$(R/((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR¢HscC stjS(N(RR&(R/((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR£LscC s t|ƒ S(N(R}(R/RA((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR¦PscC stt|ƒ|S(N(RR(R/RAR®((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR¥TscK sBddlm}t|ƒt|ttƒ|ƒdttdS(Niÿÿÿÿ(Ri(R?RR‘RR(RRARBR((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyRKXscC s)ddlj}|j|jdjƒƒS(Niÿÿÿÿi(R”R•t sinh_integralRR—(RR˜((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR—]s(RaRbRcRR¤RR¡RfR¢R£R¦R¥RKR—(((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR} s:  R~cB skeZdZeZejZed„ƒZ ed„ƒZ ed„ƒZ ed„ƒZ d„Z d„ZRS(s  Cosh integral. This function is defined for positive :math:`x` by .. math:: \operatorname{Chi}(x) = \gamma + \log{x} + \int_0^x \frac{\cosh{t} - 1}{t} \mathrm{d}t, where :math:`\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 :math:`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 cC stjS(N(RR$(R/((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR¢¬scC stjS(N(RR$(R/((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR£°scC st|ƒttS(N(R~RR(R/RA((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR¦´scC st|ƒttd|S(Ni(RœRR(R/RAR®((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR¥¸scK sCddlm}t tdt|ƒt|ttƒ|ƒdS(Niÿÿÿÿ(Ri(R?RRRR‘(RRARBR((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyRK¼scC s)ddlj}|j|jdjƒƒS(Niÿÿÿÿi(R”R•t cosh_integralRR—(RR˜((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR—Às(RaRbRcRR¤RR´R¡RfR¢R£R¦R¥RKR—(((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR~bsE  tFresnelIntegralcB sYeZdZeZed„ƒZdd„Zd„Zd„Z ed„Z ed„Z RS(s& Base class for the Fresnel integrals.cC sí|tjkrtdƒStj}|}t}|jdƒ}|dk r_| }|}t}n|jtƒ}|dk rš|jt|}|}t}n|r®|||ƒS|tj krÄtj S|ttj kré|jttj SdS(Niiÿÿÿÿ( RR(R%RXR,RlRdRt_signR$R@(R/RAtprefacttnewargtchangedRm((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR2Îs*      icC sB|dkr/|jtjt|jddƒSt||ƒ‚dS(Niii(R¤RR@RRR (RR((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyRïs #cC s|jdjS(Ni(RR<(R((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR=õscC s|j|jdjƒƒS(Ni(R9RR:(R((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR;øscK sš|jdjrI|r9t|d<|j||tjfS|tjfSn|rw|jdj||jƒ\}}n|jdjƒ\}}||fS(NiRW(RR<RXRYRR(RZ(RR[R\R_R`((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyt _as_real_imagûs (cK sÅ|jd||\}}|d |d}tj|j||t|ƒƒ|j||t|ƒƒ}|d|t|ƒ|j||t|ƒƒ|j||t|ƒƒ}||fS(NR[i(R¼RR@R9R(RR[R\R5R]R^R_R`((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyRZs ?/( RaRbRcRdReRfR2RR=R;R¼RZ(((sFlib/python2.7/site-packages/sympy/functions/special/error_functions.pyR·És!    REcB sWeZdZeZej Zee d„ƒƒZ d„Z d„Z d„Z d„ZRS(s[ Fresnel integral S. 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, gamma, expand_func >>> integrate(sin(pi*z**2/2), z) 3*fresnels(z)*gamma(3/4)/(4*gamma(7/4)) >>> expand_func(integrate(sin(pi*z**2/2), z)) fresnels(z) We can numerically evaluate the Fresnel integral to arbitrary precision on the whole complex plane: >>> fresnels(2).evalf(30) 0.343415678363698242195300815958 >>> fresnels(-2*I).evalf(30) 0.343415678363698242195300815958*I See Also ======== fresnelc: Fresnel cosine integral. References ========== .. [1] https://en.wikipedia.org/wiki/Fresnel_integral .. [2] http://dlmf.nist.gov/7 .. [3] http://mathworld.wolfram.com/FresnelIntegrals.html .. [4] http://functions.wolfram.com/GammaBetaErf/FresnelS .. [5] The converging factors for the fresnel integrals by John W. 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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, gamma, expand_func >>> integrate(cos(pi*z**2/2), z) fresnelc(z)*gamma(1/4)/(4*gamma(5/4)) >>> expand_func(integrate(cos(pi*z**2/2), z)) fresnelc(z) We can numerically evaluate the Fresnel integral to arbitrary precision on the whole complex plane: >>> fresnelc(2).evalf(30) 0.488253406075340754500223503357 >>> fresnelc(-2*I).evalf(30) -0.488253406075340754500223503357*I See Also ======== fresnels: Fresnel sine integral. References ========== .. [1] https://en.wikipedia.org/wiki/Fresnel_integral .. [2] http://dlmf.nist.gov/7 .. [3] http://mathworld.wolfram.com/FresnelIntegrals.html .. [4] http://functions.wolfram.com/GammaBetaErf/FresnelC .. [5] The converging factors for the fresnel integrals by John W. 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