The confluent hypergeometric functions \(M(a,b;z)\) and \(U(a,b;z)\) for large \(b\) and \(z\) (Q1044657)

Using a variant of Laplace's method for integrals, the authors obtain new and complete asymptotic expansions for the confluent hypergeometric functions \(M(a,b;z)\) and \(U(a,b;z)\) for large \(b\) and \(z\). For both functions the starting point is the classical integral representation and the asymptotic expansions -- which are not of Poincaré type - are different in three classes: for \(b<z+a+1\), for \(b>z+a+1\), and for \(b=z+a+1\).