A Generalized Ramanujan Master Theorem and Integral Representation of Meromorphic Functions

TL;DR

This paper extends Ramanujan's Master Theorem to meromorphic functions with arbitrary poles, providing new integral representations and opening avenues for further mathematical exploration.

Contribution

The paper generalizes Ramanujan's Master Theorem to include meromorphic functions with arbitrary pole orders, offering new integral representations and theoretical insights.

Findings

Extended theorem applies to meromorphic functions with arbitrary poles

Derived integral representations exhibit interesting properties

Potential for further research in complex analysis and Mellin transforms

Abstract

Ramanujan's Master Theorem is a decades-old theorem in the theory of Mellin transforms which has wide applications in both mathematics and high energy physics. The unconventional method of Ramanujan in his proof of the theorem left convergence issues which were later settled by Hardy. Here we extend Ramanujan's theorem to meromorphic functions with poles of arbitrary order and observe that the new theorem produces analogues of Ramanujan's famous theorem. Moreover, we find that the theorem produces integral representations for meromorphic functions which are shown to satisfy interesting properties, opening up an avenue for further study.