Whittaker functions with both parameters large: uniform approximations in terms of parabolic cylinder functions

Abstract
Synopsis: Asymptotic approximations are derived for the Whittaker functions Wκ,μ (z), Mκ, μ (z), Wικ, ιμ (iz) and Mικ, ιμ(iZ) for large positive values of the parameter μ that are uniform with respect to unrestricted values of the argument z in the open interval (0, ∞), and bounded real values of the ratio κ/μ. The approximations are in terms of parabolic cylinder functions, and in most instances are accompanied by strict error bounds.The results are derived by application of a recently-developed asymptotic theory of second-order differential equations having coalescing turning points, and an extension of the general theory of equations of this kind is also included.

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