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Uniform asymptotic approximations for the Whittaker functions ${M}_{\kappa ,i\mu }\left(z\right)$ and ${W}_{\kappa ,i\mu }\left(z\right)$. (English) Zbl 1046.33004
Uniform asymptotic approximations are obtained for the Whittaker’s confluent hypergeometric functions ${M}_{\kappa ,i\mu }\left(z\right)$ and ${W}_{\kappa ,i\mu }\left(z\right)$, where $\kappa$, $\mu$ and $z$ are real. Three cases are considered, and when taken together, result in approximations which are valid for $\kappa \to \infty$ uniformly or $0\le \mu <\infty$, $0, and also for $\mu \to \infty$ uniformly for $0\le \kappa <\infty$, $0. The results are obtained by an application of general asymptotic theories for differential equations either having a coalescing turning point and double pole with complex exponent, or a fixed simple turning point. The resulting approximations achieve a uniform reduction of free variables from three to two, and involve either modified Bessel functions or Airy functions. Explicit error bounds are available for all the approximations.
##### MSC:
 33C15 Confluent hypergeometric functions, Whittaker functions, ${}_{1}{F}_{1}$ 33C10 Bessel and Airy functions, cylinder functions, ${}_{0}{F}_{1}$ 34E20 Asymptotic singular perturbations, turning point theory, WKB methods (ODE) 41A30 Approximation by other special function classes