Closed Form Expressions for Certain Improper Integrals of Mathematical Physics

Abstract

We present new closed-form expressions for certain improper integrals of Mathematical Physics such as Ising, Box, and Associated integrals. The techniques we employ here include (a) the Method of Brackets and its modifications and suitable extensions and (b) the evaluation of the resulting Mellin-Barnes representations via the recently discovered Conic Hull method. Analytic continuations of these series solutions are then produced using the automated method of Olsson. Thus, combining all the recent advances allows for closed-form solutions for the hitherto unknown B 3 (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 is also studied. It is not yet fully resolved for the reasons we discuss in this paper.