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Series expansions of $ W\sb{k,\,m}(Z)$ involving parabolic cylinder functions


Authors: R. Wong and E. Rosenbloom
Journal: Math. Comp. 25 (1971), 783-787
MSC: Primary 33A30
DOI: https://doi.org/10.1090/S0025-5718-1971-0306566-4
MathSciNet review: 0306566
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Abstract: In this paper, an explicit error bound is obtained for an expansion of the Whittaker function, $ {W_{k,m}}(z)$, in series of parabolic cylinder functions. It is also shown that the Whittaker function may be asymptotically represented as the sum of two products where one product involves a parabolic cylinder function and the other product involves the first-order derivative of this function.


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  • [1] H. Buchholz, Die konfluente hypergeometrische Funktion mit besonderer Berücksichtigung ihrer Anwendungen, Ergebnisse der Angew. Mathematik, Band 2, Springer-Verlag, Berlin, 1953; English transl., Springer Tracts in Natural Philosophy, vol. 15, Springer-Verlag, New York, 1969. MR 14, 978; MR 39 #1692. MR 0054783 (14:978e)
  • [2] A. Erdélyi, W. Magnus, F. Oberhettinger & F. Tricomi, Higher Transcendental Functions, Vol. 2, McGraw-Hill, New York, 1953. MR 15. 419.
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  • [4] G. N. Watson, A treatise on the theory of Bessel functions, 2nd. ed., Cambridge Univ. Press, Cambridge, 1944. MR 0010746 (6:64a)
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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1971-0306566-4
Keywords: Whittaker function, parabolic cylinder functions, error bound, asymptotic representation
Article copyright: © Copyright 1971 American Mathematical Society

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