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A Unifying Integral Representation of the Gamma Function and Its Reciprocal

Peter Reinhard Hansen, Chen Tong

Abstract

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 $z\in\mathbb{C}$.

A Unifying Integral Representation of the Gamma Function and Its Reciprocal

Abstract

We derive an integral expression for the reciprocal gamma function, , that is valid for all , without the need for analytic continuation. The same integral avoids the singularities of the gamma function and satisfies for all .

Paper Structure

This paper contains 4 sections, 1 theorem, 15 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A Globally Valid Integral Expression
  3. Digamma function and the Euler-Mascheroni constant
  4. Conclusion

Key Result

Theorem 1

Let $G(z)\equiv\int_{-\infty}^{\infty}w^{1-2z}e^{w^{2}}\mathrm{d}t$, where $w=\sigma+it$ with $\sigma>0$, then

Figures (1)

  • Figure 1: The closed contour, $\mathcal{C}$, for the integral, $\oint_{\mathcal{C}}w^{-y}e^{w^{2}}\mathrm{d}w$.

Theorems & Definitions (1)

  • Theorem 1