Uniform asymptotic expansions for the Whittaker functions $M_{\kappa,\mu}(z)$ and $W_{\kappa,\mu}(z)$ with $\mu$ large

Abstract

Uniform asymptotic expansions are derived for Whittaker's confluent hypergeometric functions $M_{\kappa,\mu}(z)$ and $W_{\kappa,\mu}(z)$, as well as the numerically satisfactory companion function $W_{-\kappa,\mu}(ze^{-\pi i})$. The expansions are uniformly valid for $\mu \rightarrow \infty$, $0 \leq \kappa/\mu \leq 1-\delta <1$, and for $0 \leq \arg(z) \leq \pi$. By using appropriate connection and analytic continuation formulas these expansions can be extended to all unbounded nonzero complex $z$. The approximations come from recent asymptotic expansions involving elementary functions and Airy functions, and explicit error bounds are either provided or available.