FLINT3.7.0-dev
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acb_hypgeom_beta_lower

Synopsis#include <flint/acb_hypgeom.h>
void acb_hypgeom_beta_lower(acb_t res, const acb_t a, const acb_t b, const acb_t z, int regularized, slong prec)source

Sets res to the incomplete beta function B(a,b;z)=∫0zta−1(1−t)b−1dtB(a,b;z) = \int_0^z t^{a-1} (1-t)^{b-1} dt, or the regularized incomplete beta function I(a,b;z)=B(a,b;z)/B(a,b;1)I(a,b;z) = B(a,b;z) / B(a,b;1) if regularized is nonzero.

In general the integral is interpreted by analytic continuation. The precise definitions for all parameters are

B(a,b;z)=zaa2F1(a,1−b,a+1,z),I(a,b;z)=Γ(a+b)Γ(b)za 2F1(a,1−b,a+1,z).B(a,b;z) = \frac{z^a}{a} {}_2F_1(a, 1-b, a+1, z), \qquad I(a,b;z) = \frac{\Gamma(a+b)}{\Gamma(b)} z^a\, {}_2\mathbf{F}_1(a, 1-b, a+1, z).

With these definitions both functions are undefined for nonpositive integer aa, and II is undefined for nonpositive integer a+ba + b.

Cite this page

The FLINT team, “acb_hypgeom_beta_lower (acb_hypgeom.h),” FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev, 2026. https://flintlib.org/doc/acb_hypgeom/acb_hypgeom_beta_lower.html (generated 2026-11-02).