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-function, Euler -function, Euler integral of the first kind

A function of two variables and which, for , is defined by the equation


The values of the beta-function for various values of the parameters and are connected by the following relationships:

The following formula is valid:

If and are complex, the integral (*) converges if and . The beta-function can be expressed by the gamma-function:

How to Cite This Entry:
Beta-function. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.I. Bityutskov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article