Comment on ‘Analytical results for a Bessel function times Legendre polynomials class integrals’

A result is obtained, stemming from Gegenbauer, where the products of certain Bessel functions and exponentials are expressed in terms of an infinite series of spherical Bessel functions and products of associated Legendre functions. Closed form solutions for integrals involving Bessel functions times associated Legendre functions times exponentials, recently elucidated by Neves et al (J. Phys. A: Math. Gen. 39 L293), are then shown to result directly from the orthogonality properties of the associated Legendre functions. This result offers greater flexibility in the treatment of classical Heisenberg chains and may do so in other problems such as occur in electromagnetic diffraction theory. PACS numbers: 71.70.Gm, 73.43.Cd, 41.20.−q, 42.25.Fx, 42.60.Jf Recently, Neves et al presented a closed-form exact solution to an integral, given by equation (5), which they encountered in the solution of problems in electromagnetic vector diffraction theory [1]. Subsequently, Koumandos [2] drew attention to this integral as a special case of an integral result of Gegenbauer [3]. One of the present authors previously used a similar integral, also obtained as a special case of the same Gegenbauer integral, in treating the partition functions of magnetic particles with anisotropy [4]. More recently, the present authors have been making use of this and similar results in treating the partition functions of classical isotropic Heisenberg chains [5–7]. Historically, Podolsky and Pauling [8] presented the same result as dealt with by Neves et al. Nevertheless, the present authors share the view of Neves et al that results of this nature are of use, whilst not easily available in integral tables, and are not generally well known. A wealth of useful results, many due originally to Gegenbauer, involving associated Legendre functions are to be found in Watson’s classic treatise on Bessel functions [9]. However, these are often overlooked, perhaps owing to their expression in terms of the more general Gegenbauer polynomials, even though these are easily related to associated Legendre functions. Here we obtain a result stemming from Gegenbauer, 1751-8113/07/4614029+03$30.00 © 2007 IOP Publishing Ltd Printed in the UK 14029