Closed Form Expressions for Certain Improper Integrals of Mathematical Physics

TL;DR

This paper develops new closed-form solutions for specific improper integrals in mathematical physics, utilizing advanced techniques like the Method of Brackets, Mellin-Barnes representations, and automated hypergeometric function evaluation tools.

Contribution

It introduces a novel combination of methods to evaluate previously unknown integrals in closed form, extending the applicability of the Method of Brackets and related computational tools.

Findings

Closed-form expressions for integrals B_3(s) and B_4(s) in terms of hypergeometric functions.

Identification of complications in the Method of Brackets and proposed solutions.

Partial results for the integrals C_{5,k} with ongoing challenges.

Abstract

We present new closed-form expressions for certain improper integrals of Mathematical Physics such as certain Ising, Box, and Associated integrals. The techniques we employ here include (a) the Method of Brackets and its modifications and suitable extensions to obtain the Mellin-Barnes representation. (b) The evaluation of the resulting Mellin-Barnes representations via the recently discovered Conic Hull method via the automated package $\textit{MBConichulls.wl}$. Finally, the analytic continuations of these series solutions are then produced using the automated package \texttt{Olsson.wl}, based on the method of Olsson. Thus, combining all these recent advances allows for closed-form evaluation of the hitherto unknown $B_3(s)$, $B_4(s)$, and related integrals in terms of multivariable hypergeometric functions. Along the way, we also discuss certain complications while using the Original Method of Brackets for these evaluations and how to rectify them. The interesting cases of $C_{5,k}$ are also studied. It is not yet fully resolved for the reasons we discuss in this paper.