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.