Regularized incomplete beta function

In mathematics, the beta function, also called the of the first kind, is a special function defined by

for $Re x, Re y > 0$ .

The beta function was studied by Euler and Legendre and was given its name by Jacques Binet; its symbol $Β$ is a Greek capital beta rather than the similar Latin capital B or the Greek lowercase β.

The beta function is symmetric, meaning that

A key property of the Beta function is its relationship to the Gamma function; proof is given below in the section on relationship between gamma function and beta function

When $x$ and $y$ are positive integers, it follows from the definition of the gamma function $Γ$ that:

The Beta function satisfies several interesting identities, including

where $t \mapsto t x +$ is a truncated power function and the star denotes convolution.

The lowermost identity above shows in particular $Γ(1 / 2) = \sqrt π$ . Some of these identities, e.g. the trigonometric formula, can be applied to deriving the volume of an $n$ -ball in Cartesian coordinates.

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