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) ∫ 0 ∞ e − x / u x n j 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 implementations succeed in evaluating these integrals, the novel symbolic method implemented in this work obtains the same result and in general, it takes a fraction of the time required for the direct implementation. We test the accuracy of such analytical expressions by comparing the results with their numerical counterparts. Program summary Program title: SymbBesselJInteg Catalogue identifier: AEFY_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEFY_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 275 934 No. of bytes in distributed program, including test data, etc.: 399 705 Distribution format: tar.gz Programming language: Mathematica 7.1 Computer: Any computer running Mathematica 6.0 and later versions. Operating system: Windows Xp, Linux/Unix. RAM: 256 Mb Classification: 5. 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. Solution method: The problem is first cast into a related limit problem which can be broken into two related subproblems involving exponential and exponential integral functions. Solving the cores of each subproblem symbolically sets the stage for an involved expression tree parsing and manipulation to obtain the most simplified analytic expression for the initial problem. Running time: 1 min for typical values of the arguments and can be several mins for large values of the input variables. For the test data included, about an hour.