Kontorovich-Lebedev Integrals in Electromagnetic Scattering and Radiation Problems

Abstract

The paper explains the Kontorovich-Lebedev (KL) integral convergence issues from the viewpoint of the KL integral transformation and presents a version of the KL method which is particularly well suited for solving problems of radiation and scattering of waves with real-valued wave numbers. The approach treats the KL integrals as analytic functions of the wave number and uses the fact that the KL integrals improve their convergence as the imaginary part of the wave number grows. During the construction of a solution the wave number is assumed to be purely imaginary to ensure convergence of the involved field representations. Once the solution is constructed, the integral representation is analytically continued to a desired, for example, real, value of the wave number. The paper shows that the pole singularity in the KL integrand function is to be understood in the principal value sense. For practical evaluation of the solution, one may transform the KL integral into either a series of Bessel functions or a Sommerfeld integral. The former representation is particularly useful in the low-frequency limit, whereas the latter is ideally suited for deriving high-frequency approximations of the wave fields.