Spherical Bessel functions

TL;DR

This paper investigates indefinite integrals involving powers of x, spherical Bessel functions, and exponentials, providing conditions for smooth calculation and methods for verifying their accuracy.

Contribution

It introduces a general approach for evaluating complex integrals involving spherical Bessel functions and exponentials with exact and elementary solutions.

Findings

Derived explicit formulas for integrals involving spherical Bessel functions and exponentials.

Established conditions for smooth and accurate calculation of these integrals.

Validated the methods by measuring their equivalents and verifying accuracy.

Abstract

We examine indefinite integral involving of arbitrary power $x$, multiplied by three spherical Bessel functions of the first kind $j_{h},j_{k}$, and $j_{l}$ with integer order $h,k,l \geq 0$ and an exponential. Then we add some conditions for smooth calculation in considering the general and elementary exact evaluation. Thus, by measuring their equivalents, we can verify their accuracy