Integrals Involving a Parameter

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


Consider the integral1
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


Asymptotic Expansion Integral Representation Analytic Continuation Zeta Function Asymptotic Form 
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  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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