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Integrals Involving a Parameter

  • B. Davies
Part of the Applied Mathematical Sciences book series (AMS, volume 25)

Abstract

Consider the integral1
(1)
and suppose that we require an expansion for small values of the parameter x. When x = 0, the integral is simply a zeta function. If we attempt to find an expansion for small x by expanding the integrand in powers of x directly, the expansion will ultimately break down. To see this explicitly, suppose for simplicity that 0 < s < 1. We have then
(2)

Keywords

Asymptotic Expansion Integral Representation Analytic Continuation Zeta Function Asymptotic Form 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Footnotes

  1. 1.
    T. J. Buckholtz, and H. E. DeWitt, J. Math. Phys. (1970), 11, 477.Google Scholar
  2. 2.
    B. Davies and R. G. Storer, Phys. Rev. (1968), 171, 150.Google Scholar
  3. 3.
    These results were obtained by H. C. Levey and J. J. Mahoney, Q. Appl. Math. (1967), 26, 101, by a direct analysis. It is interesting to compare the two methods of derivation.Google Scholar

Copyright information

© Springer Science+Business Media New York 1978

Authors and Affiliations

  • B. Davies
    • 1
  1. 1.The Australian National UniversityCanberraAustralia

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