Symbolic integration of a product of two spherical Bessel functions with an additional exponential and polynomial factor

Abstract
We present a Mathematica package that performs the symbolic calculation of integrals of the form(1)underover(∫, 0, ∞) e^(- x / u)x^nj_ν(x) j_μ(x) d x where j_ν(x) and j_μ(x) denote spherical Bessel functions of integer orders, with ν ≥ 0 and μ ≥ 0. With the real parameter u > 0 and the integer n, convergence of the integral requires that n + ν + μ ≥ 0. The package provides analytical result for the integral in its most simplified form. In cases where direct Mathematica implementa...

Title of program: SymbBesselJInteg
Catalogue Id: AEFY_v1_0

Nature of problem
Integration, both analytical and numerical, of products of two spherical bessel functions with an exponential and polynomial multiplying factor can be a very complex task depending on the orders of the spherical bessel functions. The Mathematica package discussed in this paper solves this problem using a novel symbolic approach.

Versions of this program held in the CPC repository in Mendeley Data
AEFY_v1_0; SymbBesselJInteg; 10.1016/j.cpc.2010.02.006

This program has been imported from the CPC Program Library held at Queen's University Belfast (1969-2018)

摘要
本文介绍一款可完成形如式(1)的积分符号计算的Mathematica软件包：
$$int_{0}^{infty} e^{-x/u} x^n j_
u(x) j_mu(x) dx$$
其中$j_
u(x)$与$j_mu(x)$为整数阶球贝塞尔函数（spherical Bessel functions），满足$
ugeq0$、$mugeq0$。当实参数$u>0$且$n$为整数时，积分收敛的条件为$n+
u+mugeq0$。本软件包可给出该积分的最简解析形式结果。在直接通过Mathematica原生实现无法完成的场景下……

程序名称：SymbBesselJInteg
目录编号：AEFY_v1_0

待解决问题
涉及两个球贝塞尔函数与指数因子、多项式乘子的乘积的积分（包含解析积分与数值积分），根据球贝塞尔函数的阶数不同，其求解过程往往极为复杂。本文所讨论的Mathematica软件包采用新颖的符号计算方法解决了该问题。

Mendeley数据中CPC程序库存储的该程序版本：AEFY_v1_0；SymbBesselJInteg；10.1016/j.cpc.2010.02.006

本程序源自贝尔法斯特女王大学（1969-2018年）维护的CPC程序库