Symbolic integration of a product of two spherical bessel functions with an additional exponential and polynomial factor
We present a mathematica package that performs the symbolic calculation of integrals of the form \int^{\infty}_0 e^{-x/u} x^n j_ν (x) j_μ (x) dx where $j_ν (x)$ and $j_μ (x)$ denote spherical Bessel functions of integer orders, with $ν\ge 0$ and $μ\ge 0$. With the real parameter $u>0$ and the integer $n$, convergence of the integral requires that $n+ν+μ\ge 0$. 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.