Incomplete Beta Function

A generalization of the complete beta function defined by

(1)

sometimes also denoted . The so-called Chebyshev integral is given by

(2)

The incomplete beta function is implemented in the Wolfram Language as Beta[z, a, b].

It is given in terms of hypergeometric functions by

			(3)
			(4)

It is also given by the series

(5)

where  is a Pochhammer symbol.

The incomplete beta function  reduces to the usual beta function  when ,

(6)

It has derivative

(7)

and indefinite integral

(8)

REFERENCES:

Pearson, K. (Ed.). Tables of Incomplete Beta Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1968.