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)sourceSets res to the incomplete beta function B(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) if regularized is nonzero.
In general the integral is interpreted by analytic continuation. The precise definitions for all parameters are
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 a, and I is undefined for nonpositive integer a + 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 2026-10-20).
@misc{flint-doc-acb-hypgeom-acb-hypgeom-beta-lower,
author = {{The FLINT team}},
title = {{acb\_hypgeom\_beta\_lower (acb\_hypgeom.h)}},
howpublished = {FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev},
year = {2026},
url = {https://flintlib.org/doc/acb_hypgeom/acb_hypgeom_beta_lower.html},
note = {Generated 2026-10-20}
}