A generalization of the complete beta function defined by
(1)

sometimes also denoted . The socalled 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)

(8)
