Derivation of Green-type, transitional, and uniform asymptotic expansions from differential equations I. General theory, and application to modified Bessel functions of large order

Abstract

The problem of deriving Green-type asymptotic solutions from differential equations of general form d 2 y /dz 2 = X(a 2 >, z)y , for large values of a 2 , is reformulated. Combination of this formulation with the method of Mellin transforms leads further to a particularly convenient procedure for finding asymptotic expansions valid in transitional regions, and general uniform expansions. The methods are illustrated by detailed calculations for modified Bessel functions.