Uniform bounds for the complementary incomplete Gamma function

Abstract

We prove upper and lower bounds for the complementary incomplete gamma function $\G(a,z)$ with complex parameters $a$ and $z$. Our bounds are refined within the circular hyperboloid of one sheet $\{(a,z):|z|>c|a-1|\}$ with $a$ real and $z$ complex. Our results show that within the hyperboloid, $|\G(a,z)|$ is of order $|z|^{a-1}e^{-\Re(z)}$, and extends an upper estimate of Natalini and Palumbo to complex values of $z$.