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

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.