Source code for sympy.functions.special.gamma_functions

from __future__ import print_function, division

from sympy.core import Add, S, C, sympify, oo, pi
from sympy.core.function import Function, ArgumentIndexError
from .zeta_functions import zeta
from .error_functions import erf
from sympy.core import Dummy, Rational
from sympy.functions.elementary.exponential import log
from sympy.functions.elementary.integers import floor
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.combinatorial.numbers import bernoulli
from sympy.functions.combinatorial.factorials import rf
from sympy.functions.combinatorial.numbers import harmonic
from sympy.core.compatibility import xrange

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


[docs]class gamma(Function): """The gamma function returns a function which passes through the integral values of the factorial function, i.e. though defined in the complex plane, when n is an integer, `\Gamma(n) = (n - 1)!` References ========== .. [1] http://en.wikipedia.org/wiki/Gamma_function """ nargs = 1 unbranched = True def fdiff(self, argindex=1): if argindex == 1: return gamma(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 S.NaN: return S.NaN elif arg is S.Infinity: return S.Infinity elif arg.is_Integer: if arg.is_positive: return C.factorial(arg - 1) else: return S.ComplexInfinity elif arg.is_Rational: if arg.q == 2: n = abs(arg.p) // arg.q if arg.is_positive: k, coeff = n, S.One else: n = k = n + 1 if n & 1 == 0: coeff = S.One else: coeff = S.NegativeOne for i in range(3, 2*k, 2): coeff *= i if arg.is_positive: return coeff*sqrt(S.Pi) / 2**n else: return 2**n*sqrt(S.Pi) / coeff def _eval_expand_func(self, **hints): arg = self.args[0] if arg.is_Rational: if abs(arg.p) > arg.q: x = Dummy('x') n = arg.p // arg.q p = arg.p - n*arg.q return gamma(x + n)._eval_expand_func().subs(x, Rational(p, arg.q)) if arg.is_Add: coeff, tail = arg.as_coeff_add() if coeff and coeff.q != 1: intpart = floor(coeff) tail = (coeff - intpart,) + tail coeff = intpart tail = arg._new_rawargs(*tail, reeval=False) return gamma(tail)*C.RisingFactorial(tail, coeff) return self.func(*self.args) def _eval_is_real(self): return self.args[0].is_real def _eval_rewrite_as_tractable(self, z): return C.exp(loggamma(z)) def _eval_nseries(self, x, n, logx): x0 = self.args[0].limit(x, 0) if not (x0.is_Integer and x0 <= 0): return super(gamma, self)._eval_nseries(x, n, logx) t = self.args[0] - x0 return (gamma(t + 1)/rf(self.args[0], -x0 + 1))._eval_nseries(x, n, logx) def _latex(self, printer, exp=None): assert len(self.args) == 1 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. 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. See Also ======== gamma, uppergamma sympy.functions.special.hyper.hyper Examples ======== >>> from sympy import lowergamma, S >>> from sympy.abc import s, x >>> lowergamma(s, x) lowergamma(s, x) >>> lowergamma(3, x) -x**2*exp(-x) - 2*x*exp(-x) + 2 - 2*exp(-x) >>> lowergamma(-S(1)/2, x) -2*sqrt(pi)*erf(sqrt(x)) - 2*exp(-x)/sqrt(x) References ========== .. [1] Abramowitz, Milton; Stegun, Irene A., eds. (1965), Chapter 6, Section 5, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables .. [2] http://en.wikipedia.org/wiki/Incomplete_gamma_function """ nargs = 2 def fdiff(self, argindex=2): from sympy import meijerg, unpolarify if argindex == 2: a, z = self.args return C.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 sympy import unpolarify, I, factorial, exp 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 is S.One: return S.One - C.exp(-x) elif a is S.Half: 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 * C.exp(-x) if not a.is_Integer: return (cls(a + 1, x) + x**a * C.exp(-x))/a def _eval_evalf(self, prec): from sympy.mpmath import mp from sympy import Expr a = self.args[0]._to_mpmath(prec) z = self.args[1]._to_mpmath(prec) oprec = mp.prec mp.prec = prec res = mp.gammainc(a, 0, z) mp.prec = oprec return Expr._from_mpmath(res, prec) def _eval_rewrite_as_uppergamma(self, s, x): return gamma(s) - uppergamma(s, x) def _eval_rewrite_as_expint(self, s, x): from sympy 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""" 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) x**2*exp(-x) + 2*x*exp(-x) + 2*exp(-x) >>> uppergamma(-S(1)/2, x) -2*sqrt(pi)*(-erf(sqrt(x)) + 1) + 2*exp(-x)/sqrt(x) >>> uppergamma(-2, x) expint(3, x)/x**2 See Also ======== gamma, lowergamma sympy.functions.special.hyper.hyper References ========== .. [1] Abramowitz, Milton; Stegun, Irene A., eds. (1965), Chapter 6, Section 5, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables .. [2] http://en.wikipedia.org/wiki/Incomplete_gamma_function .. [3] http://en.wikipedia.org/wiki/Exponential_integral#Relation_with_other_functions """ nargs = 2 def fdiff(self, argindex=2): from sympy import meijerg, unpolarify if argindex == 2: a, z = self.args return -C.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): from sympy.mpmath import mp from sympy import Expr a = self.args[0]._to_mpmath(prec) z = self.args[1]._to_mpmath(prec) oprec = mp.prec mp.prec = prec res = mp.gammainc(a, z, mp.inf) mp.prec = oprec return Expr._from_mpmath(res, prec) @classmethod def eval(cls, a, z): from sympy import unpolarify, I, factorial, exp, expint if z.is_Number: if z is S.NaN: return S.NaN elif z is S.Infinity: return S.Zero elif z is S.Zero: return gamma(a) # We extract branching information here. C/f lowergamma. nx, n = z.extract_branch_factor() if a.is_integer and (a > 0) is True: nx = unpolarify(z) if z != nx: return uppergamma(a, nx) elif a.is_integer and (a <= 0) is True: 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 is S.One: return C.exp(-z) elif a is S.Half: 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 * C.exp(-z) elif b.is_Integer: return expint(-b, z)*unpolarify(z)**(b + 1) if not a.is_Integer: return (cls(a + 1, z) - z**a * C.exp(-z))/a def _eval_rewrite_as_lowergamma(self, s, x): return gamma(s) - lowergamma(s, x) def _eval_rewrite_as_expint(self, s, x): from sympy import expint return expint(1 - s, x)*x**s ############################################################################### ########################### GAMMA RELATED FUNCTIONS ########################### ###############################################################################
[docs]class polygamma(Function): r"""The function ``polygamma(n, z)`` returns ``log(gamma(z)).diff(n + 1)`` .. math :: \psi^{(n)}(z) = \frac{d^n}{d x^n} \log \Gamma(z) Examples ======== We can rewrite polygamma functions in terms of harmonic numbers: >>> from sympy import polygamma, harmonic, Symbol >>> x = Symbol("x") >>> 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, digamma, trigamma References ========== .. [1] http://en.wikipedia.org/wiki/Polygamma_function .. [2] http://functions.wolfram.com/GammaBetaErf/PolyGamma/ .. [3] http://functions.wolfram.com/GammaBetaErf/PolyGamma2/ """ nargs = 2 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[1].is_positive and (self.args[0] > 0) is True: return self.args[0].is_odd def _eval_is_negative(self): if self.args[1].is_positive and (self.args[0] > 0) is True: return self.args[0].is_even def _eval_is_real(self): return self.args[0].is_real def _eval_aseries(self, n, args0, x, logx): if args0[1] != oo or not \ (self.args[0].is_Integer and self.args[0].is_nonnegative): return super(polygamma, self)._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 = C.Order(1/z, x) else: m = C.ceiling((n + 1)//2) l = [bernoulli(2*k) / (2*k*z**(2*k)) for k in range(1, m)] r -= Add(*l) o = C.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 = C.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 = C.Order(1/z**(2*m), x) if n == 0: o = C.Order(1/z, x) elif n == 1: o = C.Order(1/z**2, x) r = e0._eval_nseries(z, n, logx) + o return -1 * (-1/z)**N * r @classmethod def eval(cls, n, z): n, z = list(map(sympify, (n, z))) from sympy 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 S.NaN: return S.NaN elif z is S.Infinity: if n.is_Number: if n is S.Zero: return S.Infinity else: return S.Zero elif z.is_Integer: if z.is_nonpositive: return S.ComplexInfinity else: if n is S.Zero: return -S.EulerGamma + C.harmonic(z - 1, 1) elif n.is_odd: return (-1)**(n + 1)*C.factorial(n)*zeta(n + 1, z) if n == 0 and z.is_Rational: # TODO actually *any* n/m can be done, but that is messy lookup = {S(1)/2: -2*log(2) - S.EulerGamma, S(1)/3: -S.Pi/2/sqrt(3) - 3*log(3)/2 - S.EulerGamma, S(1)/4: -S.Pi/2 - 3*log(2) - S.EulerGamma, S(3)/4: -3*log(2) - S.EulerGamma + S.Pi/2, S(2)/3: -3*log(3)/2 + S.Pi/2/sqrt(3) - S.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)]) elif 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)]) # 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(*[C.Pow( z - i, e) for i in xrange(1, int(coeff) + 1)]) else: tail = -Add(*[C.Pow( z + i, e) for i in xrange(0, int(-coeff))]) return polygamma(n, z - coeff) + (-1)**n*C.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 + C.Rational( i, coeff)) for i in xrange(0, 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 >= S.One: return (-1)**(n + 1)*C.factorial(n)*zeta(n + 1, z) else: return self def _eval_rewrite_as_harmonic(self, n, z): if n.is_integer: if n == S.Zero: return harmonic(z - 1) - S.EulerGamma else: return S.NegativeOne**(n+1) * C.factorial(n) * (C.zeta(n+1) - harmonic(z-1, n+1)) def _eval_as_leading_term(self, x): n, z = [a.as_leading_term(x) for a in self.args] o = C.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): """ The loggamma function is `\log(\Gamma(x))`. References ========== .. [1] http://mathworld.wolfram.com/LogGammaFunction.html """ nargs = 1 def _eval_nseries(self, x, n, logx=None): x0 = self.args[0].limit(x, 0) if x0 is S.Zero: f = self._eval_rewrite_as_intractable(*self.args) return f._eval_nseries(x, n, logx) return super(loggamma, self)._eval_nseries(x, n, logx) def _eval_aseries(self, n, args0, x, logx): if args0[0] != oo: return super(loggamma, self)._eval_aseries(n, args0, x, logx) z = self.args[0] m = min(n, C.ceiling((n + S(1))/2)) r = log(z)*(z - S(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 = C.Order(1, x) else: o = C.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_real(self): return self.args[0].is_real def fdiff(self, argindex=1): if argindex == 1: return polygamma(0, self.args[0]) else: raise ArgumentIndexError(self, argindex)
[docs]def digamma(x): """ The digamma function is the logarithmic derivative of the gamma function. In this case, ``digamma(x) = polygamma(0, x)``. See Also ======== gamma, trigamma, polygamma """ return polygamma(0, x)
[docs]def trigamma(x): """ The trigamma function is the second of the polygamma functions. In this case, ``trigamma(x) = polygamma(1, x)``. See Also ======== gamma, digamma, polygamma """ return polygamma(1, x)
[docs]def beta(x, y): """ Euler Beta function ``beta(x, y) == gamma(x)*gamma(y) / gamma(x+y)`` """ return gamma(x)*gamma(y) / gamma(x + y)