On infinite series concerning zeros of Bessel functions of the first kind

TL;DR

This paper presents new derivations of series involving zeros of Bessel functions using Laplace transforms and Calogero's formula, with applications demonstrated in electrical systems.

Contribution

It introduces alternative methods to derive classical identities involving Bessel zeros, expanding analytical tools for related mathematical and engineering problems.

Findings

Derived the Rayleigh-Sneddon sum using Laplace transforms.

Provided an electrical example demonstrating the sum's practical utility.

Revealed the connection between Bessel zeros and system responses.

Abstract

A relevant result independently obtained by Rayleigh and Sneddon on an identity on series involving the zeros of Bessel functions of the first kind is derived by an alternative method based on Laplace transforms. Our method leads to a Bernstein function of time, expressed by Dirichlet series, that allows us to recover the Rayleigh-Sneddon sum. We also consider another method arriving at the same result based on a relevant formula by Calogero. Moreover, we also provide an electrical example in which this sum results to be extremely useful in order to recover the analytical expression for the response of the system to a certain external input.