Source code for diofant.functions.special.beta_functions

from ...core.function import ArgumentIndexError, Function
from .gamma_functions import digamma, gamma


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# ########################## COMPLETE BETA  FUNCTION ######################## #
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[docs]class beta(Function): r""" The beta integral is called the Eulerian integral of the first kind by Legendre: .. math:: \mathrm{B}(x,y) := \int^{1}_{0} t^{x-1} (1-t)^{y-1} \mathrm{d}t. Beta function or Euler's first integral is closely associated with gamma function. The Beta function often used in probability theory and mathematical statistics. It satisfies properties like: .. math:: \mathrm{B}(a,1) = \frac{1}{a} \\ \mathrm{B}(a,b) = \mathrm{B}(b,a) \\ \mathrm{B}(a,b) = \frac{\Gamma(a) \Gamma(b)}{\Gamma(a+b)} Therefore for integral values of a and b: .. math:: \mathrm{B} = \frac{(a-1)! (b-1)!}{(a+b-1)!} Examples ======== The Beta function obeys the mirror symmetry: >>> conjugate(beta(x, y)) beta(conjugate(x), conjugate(y)) Differentiation with respect to both x and y is supported: >>> diff(beta(x, y), x) (polygamma(0, x) - polygamma(0, x + y))*beta(x, y) >>> diff(beta(x, y), y) (polygamma(0, y) - polygamma(0, x + y))*beta(x, y) We can numerically evaluate the gamma function to arbitrary precision on the whole complex plane: >>> beta(pi, pi).evalf(40) 0.02671848900111377452242355235388489324562 >>> beta(1 + I, 1 + I).evalf(20) -0.2112723729365330143 - 0.7655283165378005676*I See Also ======== diofant.functions.special.gamma_functions.gamma: Gamma function. diofant.functions.special.gamma_functions.uppergamma: Upper incomplete gamma function. diofant.functions.special.gamma_functions.lowergamma: Lower incomplete gamma function. diofant.functions.special.gamma_functions.polygamma: Polygamma function. diofant.functions.special.gamma_functions.loggamma: Log Gamma function. diofant.functions.special.gamma_functions.digamma: Digamma function. diofant.functions.special.gamma_functions.trigamma: Trigamma function. References ========== * https://en.wikipedia.org/wiki/Beta_function * http://mathworld.wolfram.com/BetaFunction.html * https://dlmf.nist.gov/5.12 """ unbranched = True def fdiff(self, argindex): x, y = self.args if argindex == 1: return beta(x, y)*(digamma(x) - digamma(x + y)) elif argindex == 2: return beta(x, y)*(digamma(y) - digamma(x + y)) else: raise ArgumentIndexError(self, argindex) @classmethod def eval(cls, x, y): pass def _eval_expand_func(self, **hints): x, y = self.args return gamma(x)*gamma(y) / gamma(x + y) def _eval_conjugate(self): return self.func(self.args[0].conjugate(), self.args[1].conjugate())