Symbolic integration of a product of two spherical Bessel functions with an additional exponential and polynomial factor
We present a Mathematica package that performs the symbolic calculation of integrals of the form integral(infinity)(0) e(-x/u)x(n) j(nu) (x) j(mu) (x) dx where J(nu) (x) and J(mu) (x) denote spherical Bessel functions of integer orders, with nu >= 0 and mu >= 0 With the real parameter u > 0 and the integer n, convergence of the integral requires that n + nu + mu >= 0 The package provides analytical result for the integral in as 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 URI. hap//cpccsqub ac uk/summaries/AEFY_v1_0 html Program obtainable from CPC Program Library, Queen's University. Belfast, N Ireland Licensing provisions Standard CPC licence, hitt) //cpc cs club ac uk/licence/licence html No of Imes 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 P rogramming language Mathematica 71 Computer Any computer running Mathematica 60 and later versions Operating system Windows X p, 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 (C) 2010 Elsevier B V All rights reserved
symbolic integration spherical bessel function