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 $z\in\mathbb{C}$.