A Unifying Integral Representation of the Gamma Function and Its Reciprocal

Peter Reinhard Hansen; Chen Tong

A Unifying Integral Representation of the Gamma Function and Its Reciprocal

Complex Variables 2026-03-05 v2 Statistics Theory Statistics Theory

Authors: Peter Reinhard Hansen , Chen Tong

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 $z\in\mathbb{C}$ .

Keywords

Gamma function integral representation reciprocal gamma analytic continuation complex analysis

Cite

@article{arxiv.2506.12112,
  title  = {A Unifying Integral Representation of the Gamma Function and Its Reciprocal},
  author = {Peter Reinhard Hansen and Chen Tong},
  journal= {arXiv preprint arXiv:2506.12112},
  year   = {2026}
}

Comments

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

A Unifying Integral Representation of the Gamma Function and Its Reciprocal

Abstract

Keywords

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