An Analytical Evaluation For The Integral Of Two Spherical Bessel Functions With An Additional Exponential And Polynomial Factor

TL;DR

This paper analytically evaluates integrals involving two spherical Bessel functions multiplied by an exponential and polynomial factor, expressing the results as finite sums over associated Legendre functions of the second kind.

Contribution

It provides a novel analytical solution for specific integrals of spherical Bessel functions with exponential and polynomial factors, expanding mathematical tools for physics applications.

Findings

Derived closed-form expressions for the integrals.

Expressed results as finite sums over associated Legendre functions.

Enhanced analytical methods for spherical Bessel function integrals.

Abstract

The integrals $\threej{\Llo}{\Llt}{\Llth} {0}{0}{0}\,\int_0^\infty \,r^{\Llth+1}\,e^{-\alpha r}\,j_\Llo(k_1r)\, j_\Llt(k_2r)\,dr$ and $\threej{\Llo}{\Llt}{\Llth} {0}{0}{0}\,\int_0^\infty \,r^{\Llth+2}\,e^{-\alpha r}\,j_\Llo(k_1r)\, j_\Llt(k_2r)\,dr$ are evaluated analytically. The result is a finite sum over the associated Legendre function of the second kind.