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

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.

Comments: 17 pages; updated references for the introduction