On the Weber integral equation and solution to the Weber-Titchmarsh problem

TL;DR

This paper establishes conditions for the existence of solutions to the Weber integral equation, solving a longstanding problem by leveraging properties of the inverse Mellin transform and related special functions.

Contribution

It provides a novel solution to the Weber-Titchmarsh problem using inverse Mellin transform techniques and integral properties of Legendre functions.

Findings

Derived sufficient conditions for Weber formal solutions

Solved the classical Weber-Titchmarsh problem

Utilized Mellin-Parseval equality and Legendre functions

Abstract

We derive sufficient conditions for the existence of the Weber formal solution of the corresponding integral equation, related to the familiar Weber-Orr integral transforms. This gives a solution to the old Weber-Titchmarsh problem (posed in Proc. Lond. Math. Soc. 22(2) (1924), pp.15, 16.) Our method involves properties of the inverse Mellin transform of integrable functions. The Mellin-Parseval equality and some integrals with the associated Legendre functions are used.