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# Beta

 Beta[a, b]gives the Euler beta function . Beta[z, a, b]gives the incomplete beta function .
• Mathematical function, suitable for both symbolic and numerical manipulation.
• .
• .
• Beta[z, a, b] has a branch cut discontinuity in the complex plane running from to .
• Beta[z0, z1, a, b] gives the generalized incomplete beta function .
• Note that the arguments in the incomplete form of Beta are arranged differently from those in the incomplete form of Gamma.
• For certain special arguments, Beta automatically evaluates to exact values.
• Beta can be evaluated to arbitrary numerical precision.
• Beta automatically threads over lists.
Exact values:
Evaluate numerically:
 Scope   (6)
Evaluate for complex arguments:
Evaluate for large arguments:
Evaluate to high precision:
The precision of the output tracks the precision of the input:
Series expansion:
Evaluate symbolically in special cases:
Infinite arguments give symbolic results:
Beta can be applied to power series:
Series expansion at poles:
Series expansion at infinity:
Evaluate symbolically at integer and half-integer orders:
Series expansion at any point:
 Applications   (4)
Plot the beta function for real positive values:
Plot of the absolute value of Beta in the complex plane:
Distribution of the average distance s of all pairs of points in a d-dimensional hypersphere:
Low-dimensional distributions can be expressed in elementary functions:
Plot distributions:
The PDF for the beta distribution for random variable :
Plot the PDF for various parameters:
Calculate the mean:
Express the Euler beta function as a ratio of Euler gamma functions:
Reduce the generalized incomplete beta function to incomplete beta functions:
Use FullSimplify to simplify beta functions:
Numerically find a root of a transcendental equation:
Sum expressions involving Beta:
Generating function:
Generate from integrals:
Obtain as special cases of hypergeometric functions:
Large arguments can give results too small to be computed explicitly:
Machine-number inputs can give high-precision results:
Algorithmically generated results often use gamma and hypergeometric rather than beta functions:
The differential equation is satisfied by a sum of incomplete beta functions:
Beta functions are typically not generated by FullSimplify:
Nest Beta over the complex plane:
The determinant of the matrix of reciprocals of beta functions is :