Source code for diofant.functions.special.beta_functions

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

###############################################################################
# ########################## COMPLETE BETA  FUNCTION ######################## #
###############################################################################

[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

========

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())