Infinite Integrals Involving Three Spherical Bessel Functions

Abstract

The integrals $\int_0^\infty {t^n j_{l_1 } (at)j_{l_2 } (bt)j_{l_3 } (ct)dt} $ are calculated, where $J_l $ is the spherical Bessel function for any integer indices n, $l_1 $, $l_2 $, $l_3 $ and real positive parameters a, b, c using two different methods. In the first $n + l_1 + l_2 + l_3 $ must be an even integer but the techniques may be generalized to ordinary Bessel functions with noninteger indices. The second method does not depend on the parity of $n + l_1 + l_2 + l_3 $ but remains valid only for integer indices.