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 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 MBConichulls.wl. Finally, the analytic continuations of these series solutions are then produced using the automated package Olsson.wl, based on the method of Olsson. Thus, combining all these recent advances allows for closed-form evaluation of the hitherto unknown B3(s), B4(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 C5,k are also studied. It is not yet fully resolved for the reasons we discuss in this paper.
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