A probabilistic proof of some integral formulas involving incomplete gamma functions

TL;DR

This paper introduces a probabilistic approach to derive closed-form integral formulas involving incomplete gamma functions, providing elementary derivations and a method that can be extended to new integrals.

Contribution

It presents a novel probabilistic proof technique for integral formulas with incomplete gamma functions, offering simpler derivations and potential for further applications.

Findings

Derived closed-form integrals involving incomplete gamma functions

Introduced a probabilistic proof method of independent interest

Potential to extend the method to new integral formulas

Abstract

The theory of normal variance mixture distributions is used to provide elementary derivations of closed-form expressions for the definite integrals $\int_0^\infty x^{-2\nu}\cos(bx)\gamma(\nu,\alpha x^2)\,\mathrm{d}x$ (for $\nu>1/2$, $b>0$ $\alpha>0$) and $\int_0^\infty x^{2\nu-1}\cos(bx)\Gamma(-\nu,\alpha x^2)\,\mathrm{d}x$ (for $\nu>0$, $b>0$ $\alpha>0$), where $\gamma(a,x)$ and $\Gamma(a,x)$ are the lower and upper incomplete gamma functions, respectively. The method of proof is of independent interest and could be used to derive further new definite integral formulas.