General Theory of Incomplete Cylindrical Functions Expressed in Poisson Form

Abstract

Basic information about the cylindrical functions of Bessel, Neumann, and Hankel was given in the preceeding chapter. These functions are all solutions of Bessel’s differential equation, and have integral representations in the Poisson and Bessel forms for which the contours of integration are completely determined. In practice, however, it frequently becomes necessary to study analogous integrals in which the contours are indeterminate. Such functions, by analogy to the incomplete elliptic integrals of Legendre or to the incomplete gamma function [12], may be called incomplete cylindrical functions.