Incomplete beta function

In mathematics, the incomplete beta function is a generalization of the beta function that replaces the definite integral of the beta function with an indefinite integral. The situation is analogous to the incomplete gamma function being a generalization of the gamma function.

The incomplete beta function is defined as

$\mathrm {B} _{x}(a,b)=\int _{0}^{x}t^{a-1}\,(1-t)^{b-1}\,\mathrm {d} t\!$

For x = 1, the incomplete beta function coincides with the complete beta function.

The regularized incomplete beta function (or regularized beta function for short) is defined in terms of the incomplete beta function and the complete beta function:

$I_{x}(a,b)={\frac {\mathrm {B} _{x}(a,b)}{\mathrm {B} (a,b)}}\!$

Properties

$I_{0}(a,b)=0\,$

$I_{1}(a,b)=1\,$

$I_{x}(a,b)=1-I_{1-x}(b,a)$

(Many other properties could be listed here.)

References

• M. Abramowitz and I. A. Stegun, eds. (1972) Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover. (See sections 6.6 and 26.5)

• W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling. (1992) Numerical Recipes in C. Cambridge, UK: Cambridge University Press. Second edition. (See section 6.4)