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 $W_{-\kappa ,\mu }(z{e}^{-\pi i})$. The expansions are uniformly valid for $\mu \to \infty$, $0\le \kappa /\mu \le 1-\delta <1$, and for $0\le \arg (z)\le \pi$. 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.

Cited by in corpus (1)

• Coulomb Green's function and an addition formula for the Whittaker functions