On an integral related to biaxially anisotropic media
In the paper under review is analyzed the following integral related to biaxially anisotropic media of theoretical electromagnetics: \[ f(x,a, b,p,m,n)= \int^\infty_0 {y^p\over y^2+x^2} J_m(ay)J_n(by)dy, \tag{*} \] where \(a\) and \(b\) (real and positive numbers) are distances from the origin to the source and observations points, \(J_m\) and \(J_n\) stand for Bessel functions and \(m\), \(n\) and \(p\) are integers with \(p+m+n\) odd. The evaluation of the integral (*) when \(b\neq a\) is developed by means of contour-integral techniques and the results are then extended to the case \(b=a\). The continuity and differentiability of the integral (*) as well as its consideration as a function of the complex variable \(x\) are also treated. In the paper the authors also discuss several relations between \(f\) and the discontinuous integrals \(WS(a,b,\lambda, \mu,\nu)\) of Weber and Schafheitlin (Mellin transform of the product of two Bessel functions \(J_\mu (ay)J_\nu(by))\) [see \textit{G. N. Watson}, A treatise on the theory of Bessel functions, 2nd. ed., Cambridge Mathematical Library. Cambridge: Cambridge Univ. Press, chapter XIII (1995; Zbl 0849.33001)]. Some generalizations are finally studied.
scientific article; zbMATH DE number 1872830
scientific article