Incomplete beta integral
real betaIncomplete (
) pure nothrow @nogc @safe;
Returns regularized incomplete beta integral of the arguments, evaluated from zero to x. The regularized incomplete beta function is defined as
betaIncomplete(a, b, x) = Γ(a + b) / ( Γ(a) Γ(b) ) * ∫0x ta-1(1-t)b-1 dt
and is the same as the cumulative distribution function of the Beta distribution.
The domain of definition is 0 <= x <= 1. In this implementation a and b are restricted to positive values. The integral from x to 1 may be obtained by the symmetry relation
betaIncompleteCompl(a, b, x ) = betaIncomplete( b, a, 1-x )
The integral is evaluated by a continued fraction expansion or, when b * x is small, by a power series.
Stephen L. Moshier (original C code). Conversion to D by Don Clugston