Computational Aspects of Incomplete Gamma Functions with Large Complex Parameters

Abstract

The incomplete gamma functions are defined by the integrals where

$$\gamma \left( {a,z} \right){\mkern 1mu} = {\mkern 1mu} \int_0^z {{t^{a - 1}}} {\mkern 1mu} {e^{ - 1}}{\mkern 1mu} dt,{\mkern 1mu} {\mkern 1mu} \Gamma \left( {a,{\mkern 1mu} z} \right){\mkern 1mu} = {\mkern 1mu} \int_z^\infty {{t^{a - 1}}} {\mkern 1mu} {e^{ - t}}{\mkern 1mu} dt$$

, where Ra > 0. We present several series representations and other relations which can be used for numerical evaluation of these functions. In particular, asymptotic representations for large complex values of the parameter a and z are considered. The results of numerical tests and the set-up of an algorithm will appear in a future paper.