FLINT3.7.0-dev
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acb_hypgeom_ei_asymp, acb_hypgeom_ei_2f2, acb_hypgeom_ei

Synopsis#include <flint/acb_hypgeom.h>
void acb_hypgeom_ei_asymp(acb_t res, const acb_t z, slong prec)source
void acb_hypgeom_ei_2f2(acb_t res, const acb_t z, slong prec)source
void acb_hypgeom_ei(acb_t res, const acb_t z, slong prec)source

Sets res to the exponential integral Ei⁡(z)\operatorname{Ei}(z), computed respectively from

Ei⁡(z)=−ezU(1,1,−z)−log⁡(−z)+12(log⁡(z)−log⁡(1z)),\operatorname{Ei}(z) = -e^z U(1,1,-z) - \log(-z) + \frac{1}{2} \left(\log(z) - \log\left(\frac{1}{z}\right) \right),
Ei⁡(z)=z 2F2(1,1;2,2;z)+γ+12(log⁡(z)−log⁡(1z)),\operatorname{Ei}(z) = z\, {}_2F_2(1, 1; 2, 2; z) + \gamma + \frac{1}{2} \left(\log(z) - \log\left(\frac{1}{z}\right) \right),

or by an automatic choice based on the expected accuracy of the two methods. The result is indeterminate if zz contains zero.

Cite this page

The FLINT team, “acb_hypgeom_ei_asymp, acb_hypgeom_ei_2f2, acb_hypgeom_ei (acb_hypgeom.h),” FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev, 2026. https://flintlib.org/doc/acb_hypgeom/acb_hypgeom_ei_asymp.html (generated 2026-11-02).