On Finite Mellin Transform via Ramanujan's Master Theorem

TL;DR

This paper develops a finite Mellin transform method using Ramanujan's Master Theorem and incomplete gamma functions, enabling new integral evaluations and validation against computational tools.

Contribution

It introduces a novel approach to construct finite Mellin transforms for functions with infinite series expansions, leveraging Ramanujan's Master Theorem.

Findings

Successfully constructs finite Mellin transforms for specific functions.

Provides new integral evaluation techniques.

Validates results with Mathematica comparisons.

Abstract

This paper aims to show that by making use of Ramanujan's Master Theorem and the properties of the lower incomplete gamma function, it is possible to construct a finite Mellin transform for the function $f(x)$ that has infinite series expansions in positive integral powers of $x$. Some applications are discussed by evaluating certain definite integrals. The obtained solutions are also compared with results from Mathematica to test the validity of the calculations.