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Explanation =========== 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 generally, $\Gamma(z)$ is defined in the whole complex plane except at the negative integers where there are simple poles. Examples ======== >>> from sympy import S, I, pi, 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 - zeta(3)/3 - 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] https://dlmf.nist.gov/5 .. [3] https://mathworld.wolfram.com/GammaFunction.html .. [4] https://functions.wolfram.com/GammaBetaErf/Gamma/ Tc|dkr<||jdtd|jdzSt||Nr5r)funcargs polygammar selfargindexs r1fdiffz gamma.fdiffrsH q==99TYq\**9Q ! +E+EE E$T844 4c~|jr0|tjur tjS|turtSt |r%|jrt |dz StjS|jr|j dkrt|j |j z}|jr|tj }}n)|dzx}}|dzdkr tj }n tj }|ttdd|zdz}|jr|t!t"zd|zz Sd|zt!t"z|z SdSdSdS)Nr5r) is_NumberrNaNrr2 is_positiver%ComplexInfinity is_RationalqabspOne NegativeOnerrangerr)clsargr0kcoeffs r1evalz gamma.evalxs@ = 5ae||u   5?-$S1W---,, 55A::CE ce+A2#$ae5 !A Aq5A::$%EEE$%MET%1Q3"2"2333E5$T"XX~144 !tDHH}u44; 5 5 5 5:r?c t|jd}|jrt|j|jkrt d}|j|jz}|j||jzz }|||z|t||jS|j rq| \}}|r%|jdkrt|}||z f|z}|}|j |ddi}||t||zS|j|jS)Nrxr5reevalF)r9rGrIrJrHrr8_eval_expand_funcsubsris_Add as_coeff_addr _new_rawargsr') r<hintsrOrTr0rJrQtailintparts r1rVzgamma._eval_expand_funcs0il ? 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Explanation =========== 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 ${}_1F_1$ 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] https://dlmf.nist.gov/8 .. [4] https://functions.wolfram.com/GammaBetaErf/Gamma2/ .. 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Explanation =========== 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 ${}_1F_1$ 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] https://dlmf.nist.gov/8 .. [4] https://functions.wolfram.com/GammaBetaErf/Gamma2/ .. [5] https://functions.wolfram.com/GammaBetaErf/Gamma3/ .. [6] https://en.wikipedia.org/wiki/Exponential_integral#Relation_with_other_functions rAc&ddlm}|dkr1|j\}}tt |  ||dz zzS|dkr?|j\}}t ||t |z|gddgdd|gg|zSt||r)rrr9rrrrr rs r1r>zuppergamma.fdiffs999999 q==9DAqA'''AE 2 2 ]]9DAqa##CFF*WWR!Q!QBPQ-R-RR R$T844 4r?cztd|jDr|jd|}|jd|}t|5t j||tj}dddn #1swxYwYtj||S|S)Nc3$K|] }|jV dSrmrrs r1rz)uppergamma._eval_evalf..rr?rr5) rr9rr*r)rinfrrrs r1rzuppergamma._eval_evalfs ..DI... . . 0 ! ''--A ! 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AaDDF ;c1"gg EQ O"%(D(D(D(D(D16q16z1B1B(D(D(D#E!E!EF\@!61"a==AQ)???|SMAFQJ7"CAFQJ>O>O5Q5Q5Q0RRRS 9  rFF7N 9 A* 88O    r?c|jd}|tjtjfvrE||jd|SdSr7rr<ros r1rbzuppergamma._eval_conjugaterr?c:t|||Srm)rr)r<rTrs r1rzuppergamma._eval_is_meromorphic"s..tQ:::r?c Bt|t||z Srm)r4rrs r1_eval_rewrite_as_lowergammaz&uppergamma._eval_rewrite_as_lowergamma%rr?c \tt|t||z Srm)rrnrrs r1rrz%uppergamma._eval_rewrite_as_tractable(s%8A;;*Q"2"222r?c 8ddlm}|d|z |||zzS)Nrrr5)rrrs r1rz"uppergamma._eval_rewrite_as_expint+s3BBBBBBva!eQ1$$r?Nr) rrrrr>rrrRrbrrrrrrr?r1rrs>>B 5 5 5 566[6pFFF ;;;+++333%%%%%r?rcveZdZdZedZdZdZdZdZ dZ dZ d Z d Z dd Zfd ZdZxZS)r:a The function ``polygamma(n, z)`` returns ``log(gamma(z)).diff(n + 1)``. Explanation =========== 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). For `n` not a nonnegative integer the generalization by Espinosa and Moll [5]_ is used: .. math:: \psi(s,z) = \frac{\zeta'(s+1, z) + (\gamma + \psi(-s)) \zeta(s+1, z)} {\Gamma(-s)} 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] https://mathworld.wolfram.com/PolygammaFunction.html .. [3] https://functions.wolfram.com/GammaBetaErf/PolyGamma/ .. [4] https://functions.wolfram.com/GammaBetaErf/PolyGamma2/ .. [5] O. Espinosa and V. Moll, "A generalized polygamma function", *Integral Transforms and Special Functions* (2004), 101-115. c6|tjus|tjur tjS|tur|jrtn tjS|jr|jr tjS|tjur*t|tdtzdz z S|jr|t us)| ttt fvrtS|jrt|dz tjz S|jrF|\}}|dkr+t%t'tj|dSdSdS|jr|jrt-|}||krt'||S|jr6tj|dzzt/|zt1|dz|zS|tjurEtj|dzzt/|zd|dzzdz zt1|dzzSdSdSdS)NrAr5F)evaluate)rrDrrrr{rdrFrLrnrrextract_multiplicativelyrr$ EulerGammarGas_numer_denomrr:reis_nonnegativerr%rr)rNr0rorJrHnzs r1rRzpolygamma.evals ::ae5L "WW.22 . \ Va. V$ $ !-  A;;QrTQ. . Y VRCxx155a88R"IEE  M!}}q|33 M''))166&yU'K'K'KLLL  M M6 \ Va. VABBww B'''| V}qs+ill:T!A#q\\IIaf}qs+ill:a!A#hqjIDQRSTQTIIUU V V V V r?cV|jdjr|jdjrdSdSdS)Nrr5T)r9rEras r1rhzpolygamma._eval_is_reals< 9Q< #  ! 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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 >>> 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).cancel() -log(x) - EulerGamma*x + pi**2*x**2/12 - x**3*zeta(3)/3 + O(x**4) We can numerically evaluate the ``loggamma`` 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] https://dlmf.nist.gov/5 .. [3] https://mathworld.wolfram.com/LogGammaFunction.html .. 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In this case, ``digamma(z) = polygamma(0, z)``. Examples ======== >>> from sympy import digamma >>> digamma(0) zoo >>> from sympy import Symbol >>> z = Symbol('z') >>> digamma(z) polygamma(0, z) To retain ``digamma`` as it is: >>> digamma(0, evaluate=False) digamma(0) >>> digamma(z, evaluate=False) digamma(z) 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] https://mathworld.wolfram.com/DigammaFunction.html .. [3] https://functions.wolfram.com/GammaBetaErf/PolyGamma2/ c|jd}t|}td||S)Nrr/r9r+r:evalfr<rronprecs r1rzdigamma._eval_evalfT9 IaLD!!A$$u$---r?r5c`|jd}td|Sr_r9r:r>r<r=ros r1r>z digamma.fdiffY' IaLA$$&&&r?cF|jd}td|jSr_r9r:rrs r1rhzdigamma._eval_is_real] IaLA&&r?cF|jd}td|jSr_r9r:rErs r1rkzdigamma._eval_is_positivea IaLA**r?cF|jd}td|jSr_r9r:rrs r1rzdigamma._eval_is_negativeerTr?c|t}tjg|z}|||||Srm)rr:rrrr<r0rrTr~ as_polygammas r1rzdigamma._eval_aseriesis;||I..  E!))!UAt<<rhrkrrrrRrVrrfrrr?r1rr$s..^... '''''''++++++=== [111...22222r?rcleZdZdZdZddZdZdZdZdZ e d Z d Z d Z d Zd ZdZdS)trigammaa^ The ``trigamma`` function is the second derivative of the ``loggamma`` function .. math:: \psi^{(1)}(z) := \frac{\mathrm{d}^{2}}{\mathrm{d} z^{2}} \log\Gamma(z). In this case, ``trigamma(z) = polygamma(1, z)``. Examples ======== >>> from sympy import trigamma >>> trigamma(0) zoo >>> from sympy import Symbol >>> z = Symbol('z') >>> trigamma(z) polygamma(1, z) To retain ``trigamma`` as it is: >>> trigamma(0, evaluate=False) trigamma(0) >>> trigamma(z, evaluate=False) trigamma(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] https://mathworld.wolfram.com/TrigammaFunction.html .. [3] https://functions.wolfram.com/GammaBetaErf/PolyGamma2/ c|jd}t|}td||S)Nrr5r/rFrHs r1rztrigamma._eval_evalfrJr?r5c`|jd}td|SryrLrMs r1r>ztrigamma.fdiffrNr?cF|jd}td|jSryrPrs r1rhztrigamma._eval_is_realrQr?cF|jd}td|jSryrSrs r1rkztrigamma._eval_is_positiverTr?cF|jd}td|jSryrVrs r1rztrigamma._eval_is_negativerTr?c|t}tjg|z}|||||Srm)rr:rrKrrXs r1rztrigamma._eval_aseriess;||I.. 5 ))!UAt<<rhrkrrrrRrVrrfrrrr?r1rmrms..^... '''''''++++++=== [111///22222r?rmc@eZdZdZdZd dZedZdZdZ dS) multigammaa The multivariate gamma function is a generalization of the gamma function .. math:: \Gamma_p(z) = \pi^{p(p-1)/4}\prod_{k=1}^p \Gamma[z + (1 - k)/2]. In a special case, ``multigamma(x, 1) = gamma(x)``. Examples ======== >>> from sympy import S, multigamma >>> from sympy import Symbol >>> x = Symbol('x') >>> p = Symbol('p', positive=True, integer=True) >>> multigamma(x, p) pi**(p*(p - 1)/4)*Product(gamma(-_k/2 + x + 1/2), (_k, 1, p)) Several special values are known: >>> multigamma(1, 1) 1 >>> multigamma(4, 1) 6 >>> multigamma(S(3)/2, 1) sqrt(pi)/2 Writing ``multigamma`` in terms of the ``gamma`` function: >>> multigamma(x, 1) gamma(x) >>> multigamma(x, 2) sqrt(pi)*gamma(x)*gamma(x - 1/2) >>> multigamma(x, 3) pi**(3/2)*gamma(x)*gamma(x - 1)*gamma(x - 1/2) Parameters ========== p : order or dimension of the multivariate gamma function See Also ======== gamma, lowergamma, uppergamma, polygamma, loggamma, digamma, trigamma, sympy.functions.special.beta_functions.beta References ========== .. 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