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_{\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.