Rayleigh and Sommerfeld representations for diffracted fields and regions of their convergence

A method is presented for determining convergence regions of the Rayleigh and Sommerfeld representations in problems of wave diffraction by cylindrical bodies of arbitrary cross section. The method is based on the analysis of the scattering pattern in the complex domain of argument values. It is shown that near an infinitely distant point the modulus of the pattern does not increase more rapidly than an exponent determined by the geometry of the body and the distribution of sources on it. The general theory is illustrated by several particular examples.