Uniform Asymptotic Expansions of Confluent Hypergeometric Functions

Abstract

Asymptotic expansions are derived for the confluent hypergeometric functions M(a, b, x) and U(a, b, x) for large b. The results are uniformly valid with respect to λ = x/b in a neighbourhood containing λ = 1; a is a fixed parameter. The expansions are derived from integral representations and contain parabolic cylinder functions and asymptotic series.