Laplace transform of spherical Bessel functions

TL;DR

This paper derives a simple elementary formula for the Laplace transform of spherical Bessel functions, revealing its polynomial structure and applying it to the memory function in a Debye model.

Contribution

It provides a new, simplified analytic expression for the Laplace transform of spherical Bessel functions, improving upon existing formulas.

Findings

Laplace transform is a polynomial in p with specific coefficients

Inverse tangent function appears as a coefficient for the highest order term

Application to the Langevin equation in a Debye model

Abstract

We provide a simple analytic formula in terms of elementary functions for the Laplace transform j_{l}(p) of the spherical Bessel function than that appearing in the literature, and we show that any such integral transform is a polynomial of order l in the variable p with constant coefficients for the first l-1 powers, and with an inverse tangent function of argument 1/p as the coefficient of the power l. We apply this formula for the Laplace transform of the memory function related to the Langevin equation in a one-dimensional Debye model.