Symbolic integration of a product of two spherical bessel functions with an additional exponential and polynomial factor

TL;DR

This paper introduces a Mathematica package that symbolically computes integrals involving products of spherical Bessel functions, an exponential, and polynomial factors, significantly speeding up calculations compared to traditional numerical methods.

Contribution

The paper presents a novel symbolic method and a Mathematica package for efficiently calculating integrals of spherical Bessel functions with exponential and polynomial factors.

Findings

The package provides analytical solutions in simplified form.

It significantly reduces computation time compared to brute-force methods.

The accuracy of the symbolic results is validated against numerical calculations.

Abstract

We present a mathematica package that performs the symbolic calculation of integrals of the form \int^{\infty}_0 e^{-x/u} x^n j_{\nu} (x) j_{\mu} (x) dx where $j_{\nu} (x)$ and $j_{\mu} (x)$ denote spherical Bessel functions of integer orders, with $\nu \ge 0$ and $\mu \ge 0$. With the real parameter $u>0$ and the integer $n$, convergence of the integral requires that $n+\nu +\mu \ge 0$. The package provides analytical result for the integral in its most simplified form. The novel symbolic method employed enables the calculation of a large number of integrals of the above form in a fraction of the time required for conventional numerical and Mathematica based brute-force methods. We test the accuracy of such analytical expressions by comparing the results with their numerical counterparts.