A Unifying Integral Representation of the Gamma Function and Its Reciprocal

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 $z\in \mathbb{C}$.

arXiv commentPlease note: The results in this manuscript have been entirely subsumed and extended by the more comprehensive framework in arXiv:2602.17007