# §8.17 Incomplete Beta Functions

## §8.17(i) Definitions and Basic Properties

Throughout §§8.17 and 8.18 we assume that , , and . However, in the case of §8.17 it is straightforward to continue most results analytically to other real values of , , and , and also to complex values.

8.17.1
8.17.2

where, as in §5.12, denotes the Beta function:

8.17.4
8.17.5 positive integers; .

Addendum: For a companion equation see (8.17.24).

8.17.6.

For a historical profile of see Dutka (1981).

## §8.17(iii) Integral Representation

With , , and ,

where and the branches of and are continuous on the path and assume their principal values when .

Further integral representations can be obtained by combining the results given in §8.17(ii) with §15.6.

With

8.17.11
8.17.12
8.17.13
8.17.15

## §8.17(v) Continued Fraction

where

The and convergents are less than , and the and convergents are greater than .