A Unifying Integral Representation of the Gamma Function and Its Reciprocal

TL;DR

This paper introduces a new integral representation for the reciprocal gamma function that is valid across the entire complex plane, avoiding singularities and eliminating the need for analytic continuation.

Contribution

It provides a unifying integral expression for the reciprocal gamma function valid for all complex numbers, simplifying analysis and computations.

Findings

Integral expression for 1/Γ(z) valid for all complex z

Expression avoids gamma function singularities

Satisfies functional equation G(1-z)=Γ(z)sin(πz)

Abstract

We derive an integral expression $G(z)$ for the reciprocal gamma function, $1/\Gamma(z)=G(z)/\pi$, that is valid for all $z\in\mathbb{C}$, without the need for analytic continuation. The same integral avoids the singularities of the gamma function and satisfies $G(1-z)=\Gamma(z)\sin(\pi z)$ for all $z\in\mathbb{C}$.