The Incomplete Beta Function and its Ratio to the Complete Beta Function
David Osborn, R. Madey · Mathematics of Computation · 1968
The incomplete beta function, ${B_x}(p,q) = \int {_0^x} {y^{p - 1}}{(1 - y)^{q - 1}}dy$, and its ratio to the complete beta function ${B_1}(p,q)$ has been calculated on an IBM-7044 computer to five significant figures and tabulated for the arguments p and q each in the interval from 0.5 to 2.0 with increments of 0.05 and for the parameter $x$ in the interval 0.1 to 1.0 with increments of 0.01. The function was evaluated by first expanding a factor of the integrand in a binomial series and then integrating each term. In order to facilitate computation, one expansion is taken for $x \leqq 1/2$ and another for $1/2 < x \leqq 1$. The expansions are truncated when the relative error is less than or equal to an arbitrarily predetermined fraction. Typical running time for achieving five significant figures on the IBM-7044 is about one minute for ${10^4}$ values.