Closed form expressions for certain improper integrals of mathematical physics
Centre for High Energy Physics, Indian Institute of Science, 560012, Bangalore, Karnataka, India
Accepted: 19 January 2024
Published online: 8 February 2024
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 , and related integrals in terms of multi-variable 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 case of is also studied. It is not yet fully resolved for the reasons we discuss in this paper.
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