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A Generalized Ramanujan Master Theorem and Integral Representation of Meromorphic Functions

Zachary P. Bradshaw, Omprakash Atale

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.

A Generalized Ramanujan Master Theorem and Integral Representation of Meromorphic Functions

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.
Paper Structure (14 theorems, 82 equations)

This paper contains 14 theorems, 82 equations.

Key Result

Theorem 1

Let $g(z)$ be analytic on the half-plane $H(\delta)=\{z\in\mathbb{C}|\textnormal{Re}(z)\ge-\delta\}$ for some $0<\delta<1$. Assume $A<\pi$ and $g$ satisfies the growth condition for all $z=v+iw\in H(\delta)$. Then holds for all $0<\textnormal{Re}(s)<\delta$.

Theorems & Definitions (29)

  • Theorem 1: Hardy-Ramanujan
  • proof
  • Theorem 2: Main Theorem
  • proof
  • Proposition 1
  • proof
  • Corollary 2.1
  • proof
  • Corollary 2.2
  • proof
  • ...and 19 more