Asymptotic and exact series representations for the incomplete Gamma function

TL;DR

The paper introduces two new series representations for the incomplete Gamma function, one asymptotic and one convergent, enhancing accuracy and applicability for various parameter values.

Contribution

It presents novel asymptotic and convergent series formulas for the incomplete Gamma function derived via a variational approach.

Findings

The asymptotic series improves upon the standard expansion.

The uniformly convergent series allows arbitrary accuracy for any parameters.

Applications demonstrate practical usefulness of the formulas.

Abstract

Using a variational approach, two new series representations for the incomplete Gamma function are derived: the first is an asymptotic series, which contains and improves over the standard asymptotic expansion; the second is a uniformly convergent series, completely analytical, which can be used to obtain arbitrarily accurate estimates of $\Gamma(a,x)$ for any value of $a$ or $x$. Applications of these formulas are discussed.