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

Published online by Cambridge University Press:  14 November 2011

F. W. J. Olver
F. W. J. Olver
Affiliation:
University of Maryland, College Park, Maryland 20742, and The National Bureau of Standards, Washington D.C. 20234
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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.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 86 , Issue 3-4 , 1980 , pp. 213 - 234
Copyright
Copyright © Royal Society of Edinburgh 1980

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References

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