New identities for the Laplace, Glasser, and Widder potential transforms and their applications

TL;DR

This paper introduces new identities for the Laplace, Glasser, and Widder potential transforms, deriving novel integral formulas and applying them to classical functions like the gamma and zeta functions, with implications for mathematical analysis.

Contribution

It presents new identities for potential transforms and demonstrates their applications to evaluate complex integrals and prove classical formulas.

Findings

New integral representations for the error function

A real analytic proof of Euler's reflection formula

Generalized integral involving the Riemann zeta function

Abstract

In this paper, we begin by applying the Laplace transform to derive closed forms for several challenging integrals that seem nearly impossible to evaluate. By utilizing the solution to the Pythagorean equation $a^2 + b^2 = c^2$, these closed forms become even more intriguing. This method allows us to provide new integral representations for the error function. Following this, we use the Fourier transform to derive formulas for the Glasser and Widder potential transforms, leading to several new and interesting corollaries. As part of the applications, we demonstrate the use of one of these integral formulas to provide a new real analytic proof of Euler's reflection formula for the gamma function. Of particular interest is a generalized integral involving the Riemann zeta function, which we also present as an application.