A Unifying Integral Representation of the Gamma Function and Its Reciprocal

We derive an integral expression $G(z)$ for the reciprocal gamma function, $1\mathrm{/}\mathrm{\Gamma }(z)=G(z)\mathrm{/}\pi$, 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 $G(1-z)=\mathrm{\Gamma }(z)\sin (\pi z)$ for all $z\in \mathbb{C}$.