Regularized integral representations of the reciprocal Gamma function

TL;DR

This paper introduces new real and complex integral representations of the reciprocal Gamma function using regularized hypersingular integrals, demonstrating their equivalence with traditional complex forms.

Contribution

It provides novel regularized integral formulas for the reciprocal Gamma function and establishes their equivalence with classical complex representations.

Findings

New real integral representation of 1/Γ(z)

Regularized complex representation along Hankel path

Equivalence with classical complex representation

Abstract

This paper establishes a real integral representation of the reciprocal $\Gamma$ function in terms of a regularized hypersingular integral. The equivalence with the usual complex representation is demonstrated. A regularized complex representation along the usual Hankel path is derived.