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A Unifying Integral Representation of the Gamma Function and Its Reciprocal
Peter Reinhard Hansen, Chen Tong · 2025-06-13 · via math.ST updates on arXiv.org

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

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