Uniform Asymptotic Expansions for Whittaker’s Confluent Hypergeometric Functions

Source abstract

The asymptotic behavior, as $\kappa \to \infty$, of the Whittaker confluent hypergeometric functions $M_{\kappa ,\mu } (z)$ and $W_{\kappa ,\mu } (z)$ is examined. Asymptotic expansions are derived in terms of Bessel and Airy functions, the results being uniformly valid for real values of $\kappa$ and $\mu$ such that ${{0 \leqq \mu } / {\kappa \leqq A < 1}}$ (A an arbitrary constant), and for all complex values of the argument z. Explicit error bounds are available for all the approximations.