FLINT3.7.0-dev
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acb_hypgeom_gamma_upper_asymp, acb_hypgeom_gamma_upper_1f1a, acb_hypgeom_gamma_upper_1f1b, acb_hypgeom_gamma_upper_singular, acb_hypgeom_gamma_upper

Synopsis#include <flint/acb_hypgeom.h>
void acb_hypgeom_gamma_upper_asymp(acb_t res, const acb_t s, const acb_t z, int regularized, slong prec)source
void acb_hypgeom_gamma_upper_1f1a(acb_t res, const acb_t s, const acb_t z, int regularized, slong prec)source
void acb_hypgeom_gamma_upper_1f1b(acb_t res, const acb_t s, const acb_t z, int regularized, slong prec)source
void acb_hypgeom_gamma_upper_singular(acb_t res, slong s, const acb_t z, int regularized, slong prec)source
void acb_hypgeom_gamma_upper(acb_t res, const acb_t s, const acb_t z, int regularized, slong prec)source

Sets res to the upper incomplete gamma function Γ(s,z)=∫z∞ts−1e−tdt\Gamma(s,z) = \int_z^\infty t^{s-1} e^{-t} dt if regularized is 0, to Q(s,z)=Γ(s,z)/Γ(s)Q(s,z) = \Gamma(s,z) / \Gamma(s) if regularized is 1, and to the generalized exponential integral z−sΓ(s,z)=E1−s(z)z^{-s} \Gamma(s,z) = E_{1-s}(z) if regularized is 2 (mainly for internal use; acb_hypgeom_expint is the interface for EsE_s).

The methods implement, in this order,

Γ(s,z)=e−zU(1−s,1−s,z),Γ(s,z)=Γ(s)−zss1F1(s,s+1,−z),\Gamma(s,z) = e^{-z} U(1-s,1-s,z), \qquad \Gamma(s,z) = \Gamma(s) - \frac{z^s}{s} {}_1F_1(s, s+1, -z),
Γ(s,z)=Γ(s)−zse−zs1F1(1,s+1,z),\Gamma(s,z) = \Gamma(s) - \frac{z^s e^{-z}}{s} {}_1F_1(1, s+1, z),
Γ(−n,z)=(−1)nn!(ψ(n+1)−log⁡(z))+(−1)n(n+1)!z 2F2(1,1;2,2+n;−z)−z−n∑k=0n−1(−z)k(k−n)k!\Gamma(-n,z) = \frac{(-1)^n}{n!} (\psi(n+1) - \log(z)) + \frac{(-1)^n}{(n+1)!} z \, {}_2F_2(1,1;2,2+n;-z) - z^{-n} \sum_{k=0}^{n-1} \frac{(-z)^k}{(k-n) k!}

for an integer n=−s≥0n = -s \ge 0 (the singular case, where the second and third formulas have removable singularities; it sums the finite sum directly and assumes that nn is not too large). The automatic version handles z=0z = 0 and s=1,2,3s = 1, 2, 3 by closed forms, returns 0 for QQ at nonpositive integer ss, uses the asymptotic expansion for large ∣z∣|z| (with a region test for complex ss and zz), the singular formula for integer s≤0s \le 0, and otherwise the first 1F1{}_1F_1 form for Re⁡z≤0\operatorname{Re} z \le 0 and the second for Re⁡z>0\operatorname{Re} z > 0. Real arguments with an inaccurate result are retried with arb_hypgeom_gamma_upper_integration.

Cite this page

The FLINT team, “acb_hypgeom_gamma_upper_asymp, acb_hypgeom_gamma_upper_1f1a, acb_hypgeom_gamma_upper_1f1b, acb_hypgeom_gamma_upper_singular, acb_hypgeom_gamma_upper (acb_hypgeom.h),” FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev, 2026. https://flintlib.org/doc/acb_hypgeom/acb_hypgeom_gamma_upper_asymp.html (generated 2026-11-02).