Source code for diofant.functions.special.gamma_functions

from mpmath import mp, workprec

from ...core import (Add, Dummy, EulerGamma, Expr, Function, I, Integer, Pow,
                     Rational, oo, pi, sympify, zoo)
from ...core.function import ArgumentIndexError
from ..combinatorial.factorials import RisingFactorial, factorial, rf
from ..combinatorial.numbers import bernoulli, harmonic
from ..elementary.exponential import exp, log
from ..elementary.integers import ceiling, floor
from ..elementary.miscellaneous import sqrt
from .error_functions import erf
from .zeta_functions import zeta


###############################################################################
# ########################## COMPLETE GAMMA FUNCTION ######################## #
###############################################################################

[docs]class gamma(Function): r""" 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 ======== Several special values are known: >>> gamma(1) 1 >>> gamma(4) 6 >>> gamma(Rational(3, 2)) sqrt(pi)/2 The Gamma function obeys the mirror symmetry: >>> conjugate(gamma(x)) gamma(conjugate(x)) Differentiation with respect to x is supported: >>> diff(gamma(x), x) gamma(x)*polygamma(0, x) Series expansion is also supported: >>> 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. diofant.functions.special.beta_functions.beta: Euler Beta function. References ========== * https://en.wikipedia.org/wiki/Gamma_function * https://dlmf.nist.gov/5 * http://mathworld.wolfram.com/GammaFunction.html * http://functions.wolfram.com/GammaBetaErf/Gamma/ """ unbranched = True def fdiff(self, argindex=1): if argindex == 1: return self.func(self.args[0])*polygamma(0, self.args[0]) else: raise ArgumentIndexError(self, argindex) @classmethod def eval(cls, arg): if arg.is_Number: if arg is oo: return oo elif arg.is_Integer: if arg.is_positive: return factorial(arg - 1) else: return zoo elif arg.is_Rational: if arg.denominator == 2: n = abs(arg.numerator) // arg.denominator if arg.is_positive: k, coeff = n, Integer(1) else: n = k = n + 1 if n & 1 == 0: coeff = Integer(1) else: coeff = Integer(-1) for i in range(3, 2*k, 2): coeff *= i if arg.is_positive: return coeff*sqrt(pi) / 2**n else: return 2**n*sqrt(pi) / coeff if arg.is_integer and arg.is_nonpositive: return zoo def _eval_expand_func(self, **hints): arg = self.args[0] if arg.is_Rational: if abs(arg.numerator) > arg.denominator: x = Dummy('x') n = arg.numerator // arg.denominator p = arg.numerator - n*arg.denominator return self.func(x + n)._eval_expand_func().subs({x: Rational(p, arg.denominator)}) if arg.is_Add: coeff, tail = arg.as_coeff_add() if coeff and coeff.denominator != 1: intpart = floor(coeff) tail = (coeff - intpart,) + tail coeff = intpart tail = arg._new_rawargs(*tail, reeval=False) return self.func(tail)*RisingFactorial(tail, coeff) return self.func(*self.args) def _eval_conjugate(self): return self.func(self.args[0].conjugate()) def _eval_is_extended_real(self): x = self.args[0] if x.is_positive or x.is_noninteger: return True def _eval_is_positive(self): x = self.args[0] if x.is_positive: return True elif x.is_noninteger: return floor(x).is_even def _eval_rewrite_as_tractable(self, z): return exp(loggamma(z)) def _eval_rewrite_as_factorial(self, z): return factorial(z - 1) def _eval_nseries(self, x, n, logx): x0 = self.args[0].limit(x, 0) if not (x0.is_Integer and x0 <= 0): return super()._eval_nseries(x, n, logx) t = self.args[0] - x0 return (self.func(t + 1)/rf(self.args[0], -x0 + 1))._eval_nseries(x, n, logx) def _latex(self, printer, exp=None): aa = printer._print(self.args[0]) if exp: return r'\Gamma^{%s}{\left(%s \right)}' % (printer._print(exp), aa) else: return r'\Gamma{\left(%s \right)}' % aa @staticmethod def _latex_no_arg(printer): return r'\Gamma'
############################################################################### # ################ LOWER and UPPER INCOMPLETE GAMMA FUNCTIONS ############### # ###############################################################################
[docs]class lowergamma(Function): r""" The lower incomplete gamma function. 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 diofant.abc import s >>> lowergamma(s, x) lowergamma(s, x) >>> lowergamma(3, x) 2 - E**(-x)*x**2 - 2*E**(-x)*x - 2*E**(-x) >>> lowergamma(-Rational(1, 2), x) -2*sqrt(pi)*erf(sqrt(x)) - 2*E**(-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. diofant.functions.special.beta_functions.beta: Euler Beta function. References ========== * https://en.wikipedia.org/wiki/Incomplete_gamma_function#Lower_incomplete_Gamma_function * Abramowitz, Milton; Stegun, Irene A., eds. (1965), Chapter 6, Section 5, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables * https://dlmf.nist.gov/8 * http://functions.wolfram.com/GammaBetaErf/Gamma2/ * http://functions.wolfram.com/GammaBetaErf/Gamma3/ """ def fdiff(self, argindex=2): from .hyper import meijerg from .. import unpolarify if argindex == 2: a, z = self.args return exp(-unpolarify(z))*z**(a - 1) elif argindex == 1: a, z = self.args return gamma(a)*digamma(a) - log(z)*uppergamma(a, z) \ - meijerg([], [1, 1], [0, 0, a], [], z) else: raise ArgumentIndexError(self, argindex) @classmethod def eval(cls, a, x): # For lack of a better place, we use this one to extract branching # information. The following can be # found in the literature (c/f references given above), albeit scattered: # 1) For fixed x != 0, lowergamma(s, x) is an entire function of s # 2) For fixed positive integers s, lowergamma(s, x) is an entire # function of x. # 3) For fixed non-positive integers s, # lowergamma(s, exp(I*2*pi*n)*x) = # 2*pi*I*n*(-1)**(-s)/factorial(-s) + lowergamma(s, x) # (this follows from lowergamma(s, x).diff(x) = x**(s-1)*exp(-x)). # 4) For fixed non-integral s, # lowergamma(s, x) = x**s*gamma(s)*lowergamma_unbranched(s, x), # where lowergamma_unbranched(s, x) is an entire function (in fact # of both s and x), i.e. # lowergamma(s, exp(2*I*pi*n)*x) = exp(2*pi*I*n*a)*lowergamma(a, x) from .. import unpolarify nx, n = x.extract_branch_factor() if a.is_integer and a.is_positive: nx = unpolarify(x) if nx != x: return lowergamma(a, nx) elif a.is_integer and a.is_nonpositive: if n != 0: return 2*pi*I*n*(-1)**(-a)/factorial(-a) + lowergamma(a, nx) elif n != 0: return exp(2*pi*I*n*a)*lowergamma(a, nx) # Special values. if a.is_Number: # TODO this should be non-recursive if a == 1: return 1 - exp(-x) elif a == Rational(1, 2): return sqrt(pi)*erf(sqrt(x)) elif a.is_Integer or (2*a).is_Integer: b = a - 1 if b.is_positive: return b*cls(b, x) - x**b * exp(-x) if not a.is_Integer: return (cls(a + 1, x) + x**a * exp(-x))/a def _eval_evalf(self, prec): a = self.args[0]._to_mpmath(prec) z = self.args[1]._to_mpmath(prec) with workprec(prec): res = mp.gammainc(a, 0, z) return Expr._from_mpmath(res, prec) def _eval_conjugate(self): z = self.args[1] if z not in (0, -oo): return self.func(self.args[0].conjugate(), z.conjugate()) def _eval_rewrite_as_uppergamma(self, s, x): return gamma(s) - uppergamma(s, x) def _eval_rewrite_as_expint(self, s, x): from .error_functions import expint if s.is_integer and s.is_nonpositive: return self return self.rewrite(uppergamma).rewrite(expint) @staticmethod def _latex_no_arg(printer): return r'\gamma'
[docs]class uppergamma(Function): r""" 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 `{}_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 diofant.abc import s >>> uppergamma(s, x) uppergamma(s, x) >>> uppergamma(3, x) E**(-x)*x**2 + 2*E**(-x)*x + 2*E**(-x) >>> uppergamma(-Rational(1, 2), x) -2*sqrt(pi)*(-erf(sqrt(x)) + 1) + 2*E**(-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. diofant.functions.special.beta_functions.beta: Euler Beta function. References ========== * https://en.wikipedia.org/wiki/Incomplete_gamma_function#Upper_incomplete_Gamma_function * Abramowitz, Milton; Stegun, Irene A., eds. (1965), Chapter 6, Section 5, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables * https://dlmf.nist.gov/8 * http://functions.wolfram.com/GammaBetaErf/Gamma2/ * http://functions.wolfram.com/GammaBetaErf/Gamma3/ * https://en.wikipedia.org/wiki/Exponential_integral#Relation_with_other_functions """ def fdiff(self, argindex=2): from .hyper import meijerg from .. import unpolarify if argindex == 2: a, z = self.args return -exp(-unpolarify(z))*z**(a - 1) elif argindex == 1: a, z = self.args return uppergamma(a, z)*log(z) + meijerg([], [1, 1], [0, 0, a], [], z) else: raise ArgumentIndexError(self, argindex) def _eval_evalf(self, prec): a = self.args[0]._to_mpmath(prec) z = self.args[1]._to_mpmath(prec) with workprec(prec): res = mp.gammainc(a, z, mp.inf) return Expr._from_mpmath(res, prec) @classmethod def eval(cls, a, z): from .error_functions import expint from .. import unpolarify if z.is_Number: if z is oo: return Integer(0) elif z == 0: # TODO: Holds only for Re(a) > 0: return gamma(a) # We extract branching information here. C/f lowergamma. nx, n = z.extract_branch_factor() if a.is_integer and a.is_positive: nx = unpolarify(z) if z != nx: return uppergamma(a, nx) elif a.is_integer and a.is_nonpositive: if n != 0: return -2*pi*I*n*(-1)**(-a)/factorial(-a) + uppergamma(a, nx) elif n != 0: return gamma(a)*(1 - exp(2*pi*I*n*a)) + exp(2*pi*I*n*a)*uppergamma(a, nx) # Special values. if a.is_Number: # TODO this should be non-recursive if a == 1: return exp(-z) elif a == Rational(1, 2): return sqrt(pi)*(1 - erf(sqrt(z))) # TODO could use erfc... elif a.is_Integer or (2*a).is_Integer: b = a - 1 if b.is_positive: return b*cls(b, z) + z**b * exp(-z) elif b.is_Integer: return expint(-b, z)*unpolarify(z)**(b + 1) else: return (cls(a + 1, z) - z**a * exp(-z))/a def _eval_conjugate(self): z = self.args[1] if z not in (0, -oo): return self.func(self.args[0].conjugate(), z.conjugate()) def _eval_rewrite_as_lowergamma(self, s, x): return gamma(s) - lowergamma(s, x) def _eval_rewrite_as_expint(self, s, x): from .error_functions import expint return expint(1 - s, x)*x**s
############################################################################### # #################### POLYGAMMA and LOGGAMMA FUNCTIONS ##################### # ###############################################################################
[docs]class polygamma(Function): r""" 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: >>> polygamma(0, 1) -EulerGamma >>> polygamma(0, Rational(1, 2)) -2*log(2) - EulerGamma >>> polygamma(0, Rational(1, 3)) -3*log(3)/2 - sqrt(3)*pi/6 - EulerGamma >>> polygamma(0, Rational(1, 4)) -3*log(2) - pi/2 - EulerGamma >>> polygamma(0, 2) -EulerGamma + 1 >>> polygamma(0, 23) -EulerGamma + 19093197/5173168 >>> 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: >>> 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) >>> 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: >>> 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. diofant.functions.special.beta_functions.beta: Euler Beta function. References ========== * https://en.wikipedia.org/wiki/Polygamma_function * http://mathworld.wolfram.com/PolygammaFunction.html * http://functions.wolfram.com/GammaBetaErf/PolyGamma/ * http://functions.wolfram.com/GammaBetaErf/PolyGamma2/ """ def fdiff(self, argindex=2): if argindex == 2: n, z = self.args[:2] return polygamma(n + 1, z) else: raise ArgumentIndexError(self, argindex) def _eval_is_positive(self): if self.args[0].is_positive and self.args[1].is_positive: if self.args[0].is_odd: return True def _eval_aseries(self, n, args0, x, logx): from ...series import Order if args0[1] != oo or not \ (self.args[0].is_Integer and self.args[0].is_nonnegative): return super()._eval_aseries(n, args0, x, logx) z = self.args[1] N = self.args[0] if N == 0: # digamma function series # Abramowitz & Stegun, p. 259, 6.3.18 r = log(z) - 1/(2*z) o = None if n < 2: o = Order(1/z, x) else: m = ceiling((n + 1)//2) l = [bernoulli(2*k) / (2*k*z**(2*k)) for k in range(1, m)] r -= Add(*l) o = Order(1/z**(2*m), x) return r._eval_nseries(x, n, logx) + o else: # proper polygamma function # Abramowitz & Stegun, p. 260, 6.4.10 # We return terms to order higher than O(x**n) on purpose # -- otherwise we would not be able to return any terms for # quite a long time! fac = gamma(N) e0 = fac + N*fac/(2*z) m = ceiling((n + 1)//2) for k in range(1, m): fac = fac*(2*k + N - 1)*(2*k + N - 2) / ((2*k)*(2*k - 1)) e0 += bernoulli(2*k)*fac/z**(2*k) o = Order(1/z**(2*m), x) if n == 0: o = Order(1/z, x) elif n == 1: o = Order(1/z**2, x) r = e0 + o return (-1 * (-1/z)**N * r)._eval_nseries(x, n, logx) @classmethod def eval(cls, n, z): n, z = list(map(sympify, (n, z))) from .. import unpolarify if n.is_integer: if n.is_nonnegative: nz = unpolarify(z) if z != nz: return polygamma(n, nz) if n == -1: return loggamma(z) else: if z.is_Number: if z is oo: if n.is_Number: if n == 0: return oo else: return Integer(0) elif z.is_Integer: if z.is_nonpositive: return zoo else: if n == 0: return -EulerGamma + harmonic(z - 1, 1) elif n.is_odd: return (-1)**(n + 1)*factorial(n)*zeta(n + 1, z) if n == 0: if z.is_Rational: # TODO actually *any* n/m can be done, but that is messy lookup = {Rational(1, 2): -2*log(2) - EulerGamma, Rational(1, 3): -pi/2/sqrt(3) - 3*log(3)/2 - EulerGamma, Rational(1, 4): -pi/2 - 3*log(2) - EulerGamma, Rational(3, 4): -3*log(2) - EulerGamma + pi/2, Rational(2, 3): -3*log(3)/2 + pi/2/sqrt(3) - EulerGamma} if z > 0: n = floor(z) z0 = z - n if z0 in lookup: return lookup[z0] + Add(*[1/(z0 + k) for k in range(n)]) else: # z < 0 n = floor(1 - z) z0 = z + n if z0 in lookup: return lookup[z0] - Add(*[1/(z0 - 1 - k) for k in range(n)]) elif z in (oo, -oo): return oo else: t = z.extract_multiplicatively(I) if t in (oo, -oo): return oo # TODO n == 1 also can do some rational z def _eval_expand_func(self, **hints): n, z = self.args if n.is_Integer and n.is_nonnegative: if z.is_Add: coeff = z.args[0] if coeff.is_Integer: e = -(n + 1) if coeff > 0: tail = Add(*[Pow(z - i, e) for i in range(1, int(coeff) + 1)]) else: tail = -Add(*[Pow(z + i, e) for i in range(int(-coeff))]) return polygamma(n, z - coeff) + (-1)**n*factorial(n)*tail elif z.is_Mul: coeff, z = z.as_two_terms() if coeff.is_Integer and coeff.is_positive: tail = [polygamma(n, z + Rational(i, coeff)) for i in range(int(coeff))] if n == 0: return Add(*tail)/coeff + log(coeff) else: return Add(*tail)/coeff**(n + 1) z *= coeff return polygamma(n, z) def _eval_rewrite_as_zeta(self, n, z): if (n - 1).is_nonnegative: return (-1)**(n + 1)*factorial(n)*zeta(n + 1, z) else: return self def _eval_rewrite_as_harmonic(self, n, z): if n.is_integer: if n == 0: return harmonic(z - 1) - EulerGamma else: return (-1)**(n+1) * factorial(n) * (zeta(n+1) - harmonic(z-1, n+1)) def _eval_as_leading_term(self, x): from ...series import Order n, z = [a.as_leading_term(x) for a in self.args] o = Order(z, x) if n == 0 and o.contains(1/x): return o.getn() * log(x) else: return self.func(n, z)
[docs]class loggamma(Function): r""" 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: >>> loggamma(-2) oo >>> loggamma(0) oo >>> loggamma(1) 0 >>> loggamma(2) 0 >>> loggamma(3) log(2) and for symbolic values: >>> n = Symbol("n", integer=True, positive=True) >>> loggamma(n) log(gamma(n)) >>> loggamma(-n) oo for half-integral values: >>> loggamma(Rational(5, 2)) log(3*sqrt(pi)/4) >>> loggamma(n/2) log(2**(-n + 1)*sqrt(pi)*gamma(n)/gamma(n/2 + 1/2)) and general rational arguments: >>> L = loggamma(Rational(16, 3)) >>> expand_func(L).doit() -5*log(3) + loggamma(1/3) + log(4) + log(7) + log(10) + log(13) >>> L = loggamma(Rational(19, 4)) >>> expand_func(L).doit() -4*log(4) + loggamma(3/4) + log(3) + log(7) + log(11) + log(15) >>> L = loggamma(Rational(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: >>> loggamma(oo) oo >>> loggamma(-oo) zoo The loggamma function obeys the mirror symmetry if `x \in \mathbb{C} \setminus \{-\infty, 0\}`: >>> c = Symbol('c', complex=True, real=False) >>> conjugate(loggamma(c)) loggamma(conjugate(c)) Differentiation with respect to x is supported: >>> diff(loggamma(x), x) polygamma(0, x) Series expansion is also supported: >>> 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: >>> 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. diofant.functions.special.beta_functions.beta: Euler Beta function. References ========== * https://en.wikipedia.org/wiki/Gamma_function * https://dlmf.nist.gov/5 * http://mathworld.wolfram.com/LogGammaFunction.html * http://functions.wolfram.com/GammaBetaErf/LogGamma/ """ @classmethod def eval(cls, z): z = sympify(z) if z.is_integer: if z.is_nonpositive: return oo elif z.is_positive: return log(gamma(z)) elif z.is_rational: p, q = z.as_numer_denom() # Half-integral values: if p.is_positive and q == 2: return log(sqrt(pi) * 2**(1 - p) * gamma(p) / gamma((p + 1) / 2)) if z is oo: return oo elif abs(z) is oo: return zoo def _eval_expand_func(self, **hints): from ...concrete import Sum z = self.args[0] if z.is_Rational: p, q = z.as_numer_denom() # General rational arguments (u + p/q) # Split z as n + p/q with p < q n = p // q p = p - n*q assert p.is_positive and q.is_positive and p < q k = Dummy("k") if n.is_positive: return loggamma(p / q) - n*log(q) + Sum(log((k - 1)*q + p), (k, 1, n)) elif n.is_negative: return loggamma(p / q) - n*log(q) + pi*I*n - Sum(log(k*q - p), (k, 1, -n)) return self def _eval_nseries(self, x, n, logx=None): x0 = self.args[0].limit(x, 0) if x0 == 0: f = self._eval_rewrite_as_intractable(*self.args) return f._eval_nseries(x, n, logx) return super()._eval_nseries(x, n, logx) def _eval_aseries(self, n, args0, x, logx): from ...series import Order if args0[0] != oo: return super()._eval_aseries(n, args0, x, logx) z = self.args[0] m = min(n, ceiling(Rational(n + 1, 2))) r = log(z)*(z - Rational(1, 2)) - z + log(2*pi)/2 l = [bernoulli(2*k) / (2*k*(2*k - 1)*z**(2*k - 1)) for k in range(1, m)] o = None if m == 0: o = Order(1, x) else: o = Order(1/z**(2*m - 1), x) # It is very inefficient to first add the order and then do the nseries return (r + Add(*l))._eval_nseries(x, n, logx) + o def _eval_rewrite_as_intractable(self, z): return log(gamma(z)) def _eval_is_extended_real(self): if self.args[0].is_nonnegative: return True def _eval_conjugate(self): z = self.args[0] if (z.is_extended_real and z.is_nonpositive) is False: return self.func(z.conjugate()) def fdiff(self, argindex=1): if argindex == 1: return polygamma(0, self.args[0]) else: raise ArgumentIndexError(self, argindex)
[docs]def digamma(x): r""" The digamma function is the first derivative of the loggamma function i.e, .. math:: \psi(x) := \frac{\mathrm{d}}{\mathrm{d} z} \log\Gamma(z) = \frac{\Gamma'(z)}{\Gamma(z) } In this case, ``digamma(z) = polygamma(0, 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. diofant.functions.special.beta_functions.beta: Euler Beta function. References ========== * https://en.wikipedia.org/wiki/Digamma_function * http://mathworld.wolfram.com/DigammaFunction.html * http://functions.wolfram.com/GammaBetaErf/PolyGamma2/ """ return polygamma(0, x)
[docs]def trigamma(x): r""" 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. diofant.functions.special.beta_functions.beta: Euler Beta function. References ========== * https://en.wikipedia.org/wiki/Trigamma_function * http://mathworld.wolfram.com/TrigammaFunction.html * http://functions.wolfram.com/GammaBetaErf/PolyGamma2/ """ return polygamma(1, x)