FLINT3.7.0-dev
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arb_hypgeom_2f1_integration

Synopsis#include <flint/arb_hypgeom.h>
void arb_hypgeom_2f1_integration(arb_t res, const arb_t a, const arb_t b, const arb_t c, const arb_t z, int regularized, slong prec)source

Computes 2F1(a,b,c,z){}_2F_1(a,b,c,z) (regularized if regularized is nonzero) by rigorous numerical integration of

2F1(a,b,c,z)=Γ(c)Γ(b)Γ(c−b)∫01tb−1(1−t)c−b−1(1−zt)−a dt.{}_2F_1(a,b,c,z) = \frac{\Gamma(c)}{\Gamma(b) \Gamma(c-b)} \int_0^1 t^{b-1} (1-t)^{c-b-1} (1-zt)^{-a} \, dt.

The result is finite only if b≥1b \ge 1, c−b≥1c - b \ge 1 and z<1z < 1, possibly after exchanging aa and bb. acb_hypgeom_2f1 uses it as a fallback for real arguments.

Cite this page

The FLINT team, “arb_hypgeom_2f1_integration (arb_hypgeom.h),” FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev, 2026. https://flintlib.org/doc/arb_hypgeom/arb_hypgeom_2f1_integration.html (generated 2026-11-02).