class_docstr: 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.
  
  Explanation
  ===========
  
  The Beta function or Euler's first integral is closely associated
  with the gamma function. The Beta function is 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)!}
  
  A special case of the Beta function when `x = y` is the
  Central Beta function. It satisfies properties like:
  
  .. math::
      \mathrm{B}(x) = 2^{1 - 2x}\mathrm{B}(x, \frac{1}{2})
      \mathrm{B}(x) = 2^{1 - 2x} cos(\pi x) \mathrm{B}(\frac{1}{2} - x, x)
      \mathrm{B}(x) = \int_{0}^{1} \frac{t^x}{(1 + t)^{2x}} dt
      \mathrm{B}(x) = \frac{2}{x} \prod_{n = 1}^{\infty} \frac{n(n + 2x)}{(n + x)^2}
  
  Examples
  ========
  
  >>> from sympy import I, pi
  >>> from sympy.abc import x, y
  
  The Beta function obeys the mirror symmetry:
  
  >>> from sympy import beta, conjugate
  >>> conjugate(beta(x, y))
  beta(conjugate(x), conjugate(y))
  
  Differentiation with respect to both $x$ and $y$ is supported:
  
  >>> from sympy import beta, diff
  >>> 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)
  
  >>> diff(beta(x), x)
  2*(polygamma(0, x) - polygamma(0, 2*x))*beta(x, x)
  
  We can numerically evaluate the Beta function to
  arbitrary precision for any complex numbers x and y:
  
  >>> from sympy import beta
  >>> beta(pi).evalf(40)
  0.02671848900111377452242355235388489324562
  
  >>> beta(1 + I).evalf(20)
  -0.2112723729365330143 - 0.7655283165378005676*I
  
  See Also
  ========
  
  gamma: Gamma function.
  uppergamma: Upper incomplete gamma function.
  lowergamma: Lower incomplete gamma function.
  polygamma: Polygamma function.
  loggamma: Log Gamma function.
  digamma: Digamma function.
  trigamma: Trigamma function.
  
  References
  ==========
  
  .. [1] https://en.wikipedia.org/wiki/Beta_function
  .. [2] https://mathworld.wolfram.com/BetaFunction.html
  .. [3] https://dlmf.nist.gov/5.12
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code: def _eval_expand_func(self, **hints):
      x, y = self.args
      return gamma(x) * gamma(y) / gamma(x + y)
  
imports: from sympy.core import S
  from sympy.core.function import Function, ArgumentIndexError
  from sympy.core.symbol import Dummy
  from sympy.functions.special.gamma_functions import gamma, digamma
  from sympy.functions.combinatorial.numbers import catalan
  from sympy.functions.elementary.complexes import conjugate
label_desc: Use image node_id 4 for calling the beta obj's underlying member method code with example usage: obj._eval_expand_func() and returns: unknown