A Unifying Integral Representation of the Gamma Function and Its Reciprocal
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866910040660639744 |
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| author | Hansen, Peter Reinhard Tong, Chen |
| author_facet | Hansen, Peter Reinhard Tong, Chen |
| contents | 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 $z\in\mathbb{C}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_12112 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Unifying Integral Representation of the Gamma Function and Its Reciprocal Hansen, Peter Reinhard Tong, Chen Complex Variables Statistics Theory 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 $z\in\mathbb{C}$. |
| title | A Unifying Integral Representation of the Gamma Function and Its Reciprocal |
| topic | Complex Variables Statistics Theory |
| url | https://arxiv.org/abs/2506.12112 |