An Analytical Evaluation For The Integral Of Two Spherical Bessel Functions With An Additional Exponential And Polynomial Factor

Abstract

The integrals $\threej{\Llo}{\Llt}{\Llth} {0}{0}{0}\,\int_0^\infty \,r^{\Llth+1}\,e^{-\alpha r}\,j_\Llo(k_1r)\, j_\Llt(k_2r)\,dr$ and $\threej{\Llo}{\Llt}{\Llth} {0}{0}{0}\,\int_0^\infty \,r^{\Llth+2}\,e^{-\alpha r}\,j_\Llo(k_1r)\, j_\Llt(k_2r)\,dr$ are evaluated analytically. The result is a finite sum over the associated Legendre function of the second kind.