Calculations of integrals of products of Bessel functions

Abstract

where n\, n2, ns are zero or positive integers, n + n2 + ns is even and a, b are real positive numbers. The symbols Ui are used for convenient reference. Integrals (1.1) and (1.2) together with integrals of threefold and fourfold products of associated Legendre functions [7], were used in the calculation of virial coefficients in statistical mechanics [3], [4], [5], [6]. The usual numerical integration techniques such as Simpson's method, Gauss' method, method of indefinite integral of polynomials [7] etc., when applied to integrals with oscillating integrands such as in (1.1) and (1.2) are inefficient. Values of integrals (1.1) and (1.2) can be obtained by transforming them into Mellin-Barnes integrals [1], [2], etc., or Meijer's G-functions [2], and application of the residue calculus as developed by one of the authors in [3] leads to the exact determination of the integrals. As the Mellin-Barnes integrands are rather complicated, one has to do considerable scanning to determine the actual poles. In this paper, the scanning process and the evaluation of the residues at these poles is computer programmed.