ó ¡¼™\c@ sÄddlmZmZddlmZmZmZmZmZm Z m Z ddl m Z ddl mZmZddlmZddlmZddlmZdd lmZmZdd lmZmZdd lmZmZdd lm Z dd l!m"Z"m#Z#m$Z$ddl%m&Z&m'Z'ddl(m)Z)m*Z*m+Z+defd„ƒYZ,defd„ƒYZ-defd„ƒYZ.defd„ƒYZ/defd„ƒYZ0d„Z1d„Z2dS(iÿÿÿÿ(tprint_functiontdivision(tAddtStsympifytootpitDummyt expand_func(trange(tFunctiontArgumentIndexError(tRational(tPowi(tzeta(terfterfc(texptlog(tceilingtfloor(tsqrt(tsintcostcot(t bernoullitharmonic(t factorialtrftRisingFactorialtgammacB sweZdZeZdd„Zed„ƒZd„Zd„Z d„Z d„Z d„Z d „Z d „Zd „ZRS( sÊ The gamma function .. math:: \Gamma(x) := \int^{\infty}_{0} t^{x-1} e^{-t} \mathrm{d}t. The ``gamma`` function implements the function which passes through the values of the factorial function, i.e. `\Gamma(n) = (n - 1)!` when n is an integer. More general, `\Gamma(z)` is defined in the whole complex plane except at the negative integers where there are simple poles. Examples ======== >>> from sympy import S, I, pi, oo, gamma >>> from sympy.abc import x Several special values are known: >>> gamma(1) 1 >>> gamma(4) 6 >>> gamma(S(3)/2) sqrt(pi)/2 The Gamma function obeys the mirror symmetry: >>> from sympy import conjugate >>> conjugate(gamma(x)) gamma(conjugate(x)) Differentiation with respect to x is supported: >>> from sympy import diff >>> diff(gamma(x), x) gamma(x)*polygamma(0, x) Series expansion is also supported: >>> from sympy import series >>> series(gamma(x), x, 0, 3) 1/x - EulerGamma + x*(EulerGamma**2/2 + pi**2/12) + x**2*(-EulerGamma*pi**2/12 + polygamma(2, 1)/6 - EulerGamma**3/6) + O(x**3) We can numerically evaluate the gamma function to arbitrary precision on the whole complex plane: >>> gamma(pi).evalf(40) 2.288037795340032417959588909060233922890 >>> gamma(1+I).evalf(20) 0.49801566811835604271 - 0.15494982830181068512*I See Also ======== lowergamma: Lower incomplete gamma function. uppergamma: Upper incomplete gamma function. polygamma: Polygamma function. loggamma: Log Gamma function. digamma: Digamma function. trigamma: Trigamma function. sympy.functions.special.beta_functions.beta: Euler Beta function. References ========== .. [1] https://en.wikipedia.org/wiki/Gamma_function .. [2] http://dlmf.nist.gov/5 .. [3] http://mathworld.wolfram.com/GammaFunction.html .. 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It can be defined as the meromorphic continuation of .. math:: \gamma(s, x) := \int_0^x t^{s-1} e^{-t} \mathrm{d}t = \Gamma(s) - \Gamma(s, x). This can be shown to be the same as .. math:: \gamma(s, x) = \frac{x^s}{s} {}_1F_1\left({s \atop s+1} \middle| -x\right), where :math:`{}_1F_1` is the (confluent) hypergeometric function. Examples ======== >>> from sympy import lowergamma, S >>> from sympy.abc import s, x >>> lowergamma(s, x) lowergamma(s, x) >>> lowergamma(3, x) -2*(x**2/2 + x + 1)*exp(-x) + 2 >>> lowergamma(-S(1)/2, x) -2*sqrt(pi)*erf(sqrt(x)) - 2*exp(-x)/sqrt(x) See Also ======== gamma: Gamma function. uppergamma: Upper incomplete gamma function. polygamma: Polygamma function. loggamma: Log Gamma function. digamma: Digamma function. trigamma: Trigamma function. sympy.functions.special.beta_functions.beta: Euler Beta function. References ========== .. [1] https://en.wikipedia.org/wiki/Incomplete_gamma_function#Lower_incomplete_Gamma_function .. [2] Abramowitz, Milton; Stegun, Irene A., eds. (1965), Chapter 6, Section 5, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables .. [3] http://dlmf.nist.gov/8 .. [4] http://functions.wolfram.com/GammaBetaErf/Gamma2/ .. [5] http://functions.wolfram.com/GammaBetaErf/Gamma3/ icC sÌddlm}m}|dkrN|j\}}t||ƒ ƒ||dS|dkr¹|j\}}t|ƒt|ƒt|ƒt||ƒ|gddgdd|gg|ƒSt ||ƒ‚dS(Niÿÿÿÿ(tmeijergt unpolarifyiii( tsympyR`RaR RRtdigammaRt uppergammaR (R"R#R`RataRL((sFlib/python2.7/site-packages/sympy/functions/special/gamma_functions.pyR$õs  Pc C sñddlm}m}|dkr)tjS|jƒ\}}|jru|jru||ƒ}||krút||ƒSn…|jrÇ|j rÇ|dkrúdt ||d| t | ƒt||ƒSn3|dkrút dt |||ƒt||ƒS|j rí|tjkr$tjt | ƒS|tjkrMtt ƒtt|ƒƒS|jscd|jrí|d}|jrJ|jrÏt |ƒt | ƒt |ƒtgt|ƒD]}||t |ƒ^q¬ŒSt|ƒttj|ƒtt ƒt | ƒtgtd|tjƒD](}||tjttj|ƒ^qŒSn|jsêdtj|t tt|ƒƒtd|ƒt | ƒtgtdtdƒd|ƒD]0}|||dt|ƒt||ƒ^q®ŒSqíndS(Niÿÿÿÿ(RatIiiii(RbRaRfRtZerotextract_branch_factort is_integerR)R_tis_nonpositiveRRRR%R/tHalfRRR(RR R( R2ReR9RaRftnxR4tbR5((sFlib/python2.7/site-packages/sympy/functions/special/gamma_functions.pyR8s4    4 '    P{ cC s«ddlm}m}ddlm}td„|jDƒƒr£|jdj|ƒ}|jdj|ƒ}||ƒ|j|d|ƒ}WdQX|j ||ƒS|SdS(Niÿÿÿÿ(tmptworkprec(tExprcs s|]}|jVqdS(N(t is_number(t.0R9((sFlib/python2.7/site-packages/sympy/functions/special/gamma_functions.pys 5sii( tmpmathRnRoRbRpRWR t _to_mpmathtgammainct _from_mpmath(R"tprecRnRoRpReRLtres((sFlib/python2.7/site-packages/sympy/functions/special/gamma_functions.pyt _eval_evalf2s cC sL|jd}|tjtjfkrH|j|jdjƒ|jƒƒSdS(Nii(R RRgtNegativeInfinityRRD(R"RL((sFlib/python2.7/site-packages/sympy/functions/special/gamma_functions.pyRE>s cK st|ƒt||ƒS(N(RRd(R"tsR9RM((sFlib/python2.7/site-packages/sympy/functions/special/gamma_functions.pyt_eval_rewrite_as_uppergammaCscK s<ddlm}|jr&|jr&|S|jtƒj|ƒS(Niÿÿÿÿ(texpint(RbR}RiRjtrewriteRd(R"R{R9RMR}((sFlib/python2.7/site-packages/sympy/functions/special/gamma_functions.pyt_eval_rewrite_as_expintFs( RZR[R\R$R^R8RyRER|R(((sFlib/python2.7/site-packages/sympy/functions/special/gamma_functions.pyR_Âs0 0  RdcB sVeZdZdd„Zd„Zed„ƒZd„Zd„Zd„Z d„Z RS( s¡ The upper incomplete gamma function. It can be defined as the meromorphic continuation of .. math:: \Gamma(s, x) := \int_x^\infty t^{s-1} e^{-t} \mathrm{d}t = \Gamma(s) - \gamma(s, x). where `\gamma(s, x)` is the lower incomplete gamma function, :class:`lowergamma`. This can be shown to be the same as .. math:: \Gamma(s, x) = \Gamma(s) - \frac{x^s}{s} {}_1F_1\left({s \atop s+1} \middle| -x\right), where :math:`{}_1F_1` is the (confluent) hypergeometric function. The upper incomplete gamma function is also essentially equivalent to the generalized exponential integral: .. math:: \operatorname{E}_{n}(x) = \int_{1}^{\infty}{\frac{e^{-xt}}{t^n} \, dt} = x^{n-1}\Gamma(1-n,x). Examples ======== >>> from sympy import uppergamma, S >>> from sympy.abc import s, x >>> uppergamma(s, x) uppergamma(s, x) >>> uppergamma(3, x) 2*(x**2/2 + x + 1)*exp(-x) >>> uppergamma(-S(1)/2, x) -2*sqrt(pi)*erfc(sqrt(x)) + 2*exp(-x)/sqrt(x) >>> uppergamma(-2, x) expint(3, x)/x**2 See Also ======== gamma: Gamma function. lowergamma: Lower incomplete gamma function. polygamma: Polygamma function. loggamma: Log Gamma function. digamma: Digamma function. trigamma: Trigamma function. sympy.functions.special.beta_functions.beta: Euler Beta function. References ========== .. [1] https://en.wikipedia.org/wiki/Incomplete_gamma_function#Upper_incomplete_Gamma_function .. [2] Abramowitz, Milton; Stegun, Irene A., eds. (1965), Chapter 6, Section 5, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables .. [3] http://dlmf.nist.gov/8 .. [4] http://functions.wolfram.com/GammaBetaErf/Gamma2/ .. [5] http://functions.wolfram.com/GammaBetaErf/Gamma3/ .. [6] https://en.wikipedia.org/wiki/Exponential_integral#Relation_with_other_functions icC s¹ddlm}m}|dkrO|j\}}t||ƒ ƒ ||dS|dkr¦|j\}}t||ƒt|ƒ|gddgdd|gg|ƒSt||ƒ‚dS(Niÿÿÿÿ(R`Raiii(RbR`RaR RRdRR (R"R#R`RaReRL((sFlib/python2.7/site-packages/sympy/functions/special/gamma_functions.pyR$Šs  <cC sªddlm}m}ddlm}td„|jDƒƒr¦|jdj|ƒ}|jdj|ƒ}||ƒ|j|||j ƒ}WdQX|j ||ƒS|S(Niÿÿÿÿ(RnRo(Rpcs s|]}|jVqdS(N(Rq(RrR9((sFlib/python2.7/site-packages/sympy/functions/special/gamma_functions.pys ˜sii( RsRnRoRbRpRWR RtRutinfRv(R"RwRnRoRpReRLRx((sFlib/python2.7/site-packages/sympy/functions/special/gamma_functions.pyRy•s c C sddlm}m}m}|jrm|tjkr;tjS|tjkrQtjS|tjkrmt |ƒSn|j ƒ\}}|j rÂ|dkt krÂ||ƒ}||krxt ||ƒSn¶|j r|dkt kr|dkrxdt||d| t| ƒt ||ƒSn[|dkrxt |ƒdtdt|||ƒtdt|||ƒt ||ƒS|jr™|tjkr›t| ƒS|tjkrÄttƒtt|ƒƒS|jsÚd|jr™|d}|jrÓ|j r<t| ƒt|ƒtgt|ƒD]} || t| ƒ^qŒSt |ƒtt|ƒƒd|tdƒdt| ƒt|ƒtgt|tjƒD]1} t tj | ƒ| | t d|ƒ^q–ŒSn)|jrü|| |ƒ||ƒ|dS|js–dtj|ttt|ƒƒt d|ƒ||t| ƒtgttj|ƒD],} || t |ƒt || dƒ^q^ŒSq™ndS(Niÿÿÿÿ(RaRfR}iiþÿÿÿiii(RbRaRfR}R%RR&R'RgRRhRiRGRdRRRR/RkRRR(R)RR ( R2ReRLRaRfR}RlR4RmR5((sFlib/python2.7/site-packages/sympy/functions/special/gamma_functions.pyR8 sB     4 O     F—   cC sL|jd}|tjtjfkrH|j|jdjƒ|jƒƒSdS(Nii(R RRgRzRRD(R"RL((sFlib/python2.7/site-packages/sympy/functions/special/gamma_functions.pyREËs cK st|ƒt||ƒS(N(RR_(R"R{R9RM((sFlib/python2.7/site-packages/sympy/functions/special/gamma_functions.pyt_eval_rewrite_as_lowergammaÐscK s)ddlm}|d||ƒ||S(Niÿÿÿÿ(R}i(RbR}(R"R{R9RMR}((sFlib/python2.7/site-packages/sympy/functions/special/gamma_functions.pyRÓscC s9ddlj}|j|jdjƒ|jdjƒƒS(Niÿÿÿÿii(RVRWRR RX(R"RY((sFlib/python2.7/site-packages/sympy/functions/special/gamma_functions.pyRX×s( RZR[R\R$RyR^R8RERRRX(((sFlib/python2.7/site-packages/sympy/functions/special/gamma_functions.pyRdMs: +   R!cB sqeZdZdd„Zd„Zd„Zd„Zd„Zed„ƒZ d„Z d „Z d „Z d „Z RS( s' The function ``polygamma(n, z)`` returns ``log(gamma(z)).diff(n + 1)``. It is a meromorphic function on `\mathbb{C}` and defined as the (n+1)-th derivative of the logarithm of the gamma function: .. math:: \psi^{(n)} (z) := \frac{\mathrm{d}^{n+1}}{\mathrm{d} z^{n+1}} \log\Gamma(z). Examples ======== Several special values are known: >>> from sympy import S, polygamma >>> polygamma(0, 1) -EulerGamma >>> polygamma(0, 1/S(2)) -2*log(2) - EulerGamma >>> polygamma(0, 1/S(3)) -log(3) - sqrt(3)*pi/6 - EulerGamma - log(sqrt(3)) >>> polygamma(0, 1/S(4)) -pi/2 - log(4) - log(2) - EulerGamma >>> polygamma(0, 2) 1 - EulerGamma >>> polygamma(0, 23) 19093197/5173168 - EulerGamma >>> from sympy import oo, I >>> polygamma(0, oo) oo >>> polygamma(0, -oo) oo >>> polygamma(0, I*oo) oo >>> polygamma(0, -I*oo) oo Differentiation with respect to x is supported: >>> from sympy import Symbol, diff >>> x = Symbol("x") >>> diff(polygamma(0, x), x) polygamma(1, x) >>> diff(polygamma(0, x), x, 2) polygamma(2, x) >>> diff(polygamma(0, x), x, 3) polygamma(3, x) >>> diff(polygamma(1, x), x) polygamma(2, x) >>> diff(polygamma(1, x), x, 2) polygamma(3, x) >>> diff(polygamma(2, x), x) polygamma(3, x) >>> diff(polygamma(2, x), x, 2) polygamma(4, x) >>> n = Symbol("n") >>> diff(polygamma(n, x), x) polygamma(n + 1, x) >>> diff(polygamma(n, x), x, 2) polygamma(n + 2, x) We can rewrite polygamma functions in terms of harmonic numbers: >>> from sympy import harmonic >>> polygamma(0, x).rewrite(harmonic) harmonic(x - 1) - EulerGamma >>> polygamma(2, x).rewrite(harmonic) 2*harmonic(x - 1, 3) - 2*zeta(3) >>> ni = Symbol("n", integer=True) >>> polygamma(ni, x).rewrite(harmonic) (-1)**(n + 1)*(-harmonic(x - 1, n + 1) + zeta(n + 1))*factorial(n) See Also ======== gamma: Gamma function. lowergamma: Lower incomplete gamma function. uppergamma: Upper incomplete gamma function. loggamma: Log Gamma function. digamma: Digamma function. trigamma: Trigamma function. sympy.functions.special.beta_functions.beta: Euler Beta function. References ========== .. [1] https://en.wikipedia.org/wiki/Polygamma_function .. [2] http://mathworld.wolfram.com/PolygammaFunction.html .. [3] http://functions.wolfram.com/GammaBetaErf/PolyGamma/ .. 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RKcB skeZdZed„ƒZd„Zd d„Zd„Zd„Z d„Z d„Z dd „Z d „Z RS( sà The ``loggamma`` function implements the logarithm of the gamma function i.e, `\log\Gamma(x)`. Examples ======== Several special values are known. For numerical integral arguments we have: >>> from sympy import loggamma >>> loggamma(-2) oo >>> loggamma(0) oo >>> loggamma(1) 0 >>> loggamma(2) 0 >>> loggamma(3) log(2) and for symbolic values: >>> from sympy import Symbol >>> n = Symbol("n", integer=True, positive=True) >>> loggamma(n) log(gamma(n)) >>> loggamma(-n) oo for half-integral values: >>> from sympy import S, pi >>> loggamma(S(5)/2) log(3*sqrt(pi)/4) >>> loggamma(n/2) log(2**(1 - n)*sqrt(pi)*gamma(n)/gamma(n/2 + 1/2)) and general rational arguments: >>> from sympy import expand_func >>> L = loggamma(S(16)/3) >>> expand_func(L).doit() -5*log(3) + loggamma(1/3) + log(4) + log(7) + log(10) + log(13) >>> L = loggamma(S(19)/4) >>> expand_func(L).doit() -4*log(4) + loggamma(3/4) + log(3) + log(7) + log(11) + log(15) >>> L = loggamma(S(23)/7) >>> expand_func(L).doit() -3*log(7) + log(2) + loggamma(2/7) + log(9) + log(16) The loggamma function has the following limits towards infinity: >>> from sympy import oo >>> loggamma(oo) oo >>> loggamma(-oo) zoo The loggamma function obeys the mirror symmetry if `x \in \mathbb{C} \setminus \{-\infty, 0\}`: >>> from sympy.abc import x >>> from sympy import conjugate >>> conjugate(loggamma(x)) loggamma(conjugate(x)) Differentiation with respect to x is supported: >>> from sympy import diff >>> diff(loggamma(x), x) polygamma(0, x) Series expansion is also supported: >>> from sympy import series >>> series(loggamma(x), x, 0, 4) -log(x) - EulerGamma*x + pi**2*x**2/12 + x**3*polygamma(2, 1)/6 + O(x**4) We can numerically evaluate the gamma function to arbitrary precision on the whole complex plane: >>> from sympy import I >>> loggamma(5).evalf(30) 3.17805383034794561964694160130 >>> loggamma(I).evalf(20) -0.65092319930185633889 - 1.8724366472624298171*I See Also ======== gamma: Gamma function. lowergamma: Lower incomplete gamma function. uppergamma: Upper incomplete gamma function. polygamma: Polygamma function. digamma: Digamma function. trigamma: Trigamma function. sympy.functions.special.beta_functions.beta: Euler Beta function. References ========== .. [1] https://en.wikipedia.org/wiki/Gamma_function .. [2] http://dlmf.nist.gov/5 .. [3] http://mathworld.wolfram.com/LogGammaFunction.html .. 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See Also ======== gamma: Gamma function. lowergamma: Lower incomplete gamma function. uppergamma: Upper incomplete gamma function. polygamma: Polygamma function. loggamma: Log Gamma function. trigamma: Trigamma function. sympy.functions.special.beta_functions.beta: Euler Beta function. References ========== .. [1] https://en.wikipedia.org/wiki/Digamma_function .. [2] http://mathworld.wolfram.com/DigammaFunction.html .. [3] http://functions.wolfram.com/GammaBetaErf/PolyGamma2/ i(R!(R9((sFlib/python2.7/site-packages/sympy/functions/special/gamma_functions.pyRc¹scC s td|ƒS(s The trigamma function is the second derivative of the loggamma function i.e, .. math:: \psi^{(1)}(z) := \frac{\mathrm{d}^{2}}{\mathrm{d} z^{2}} \log\Gamma(z). In this case, ``trigamma(z) = polygamma(1, z)``. See Also ======== gamma: Gamma function. lowergamma: Lower incomplete gamma function. uppergamma: Upper incomplete gamma function. polygamma: Polygamma function. loggamma: Log Gamma function. digamma: Digamma function. sympy.functions.special.beta_functions.beta: Euler Beta function. References ========== .. [1] https://en.wikipedia.org/wiki/Trigamma_function .. [2] http://mathworld.wolfram.com/TrigammaFunction.html .. 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