The singularly perturbed Bessel equation in complex domains

Abstract

We use the method of regularization to construct two kinds of regularized asymptotic expansions (in a complex parameter) for a fundamental system of solutions of the Bessel equation. Expansions of the first kind are defined in the closed complex plane of the independent variable except for singular points of the spectral functions of the initial operator. We determine the domains of uniform and non-uniform convergence of the series involved. We study the resulting formulae on the positive real axis and prove that they yield Debye's familiar asymptotic expansions for Bessel functions on the interval (0,1), which lies in the domain of non-uniform convergence. The second kind of regularized uniform asymptotic expansions is constructed near a regular singular point in another domain of values of the parameter in the equations. Using these results, we get uniform asymptotic expansions of solutions of a boundary-value problem for the non-homogenous and homogeneous Bessel equations.