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for Re(x), Re(y) > 0.
The beta function is symmetric, meaning that
It has many other forms, including:
The derivatives follow:
where is the digamma function.
The Nörlund-Rice integral is a contour integral involving the beta function.
Incomplete beta functionEdit
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
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:
(Many other properties could be listed here.)
- Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. (See §6.2, 6.6, and 26.5)
- W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling. Numerical Recipes in C. Cambridge, UK: Cambridge University Press, 1992. Second edition. (See section 6.4)