Integrals Involving a Parameter
Part of the Applied Mathematical Sciences book series (AMS, volume 25)
Consider the integral1and 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
KeywordsAsymptotic Expansion Integral Representation Analytic Continuation Zeta Function Asymptotic Form
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- 1.T. J. Buckholtz, and H. E. DeWitt, J. Math. Phys. (1970), 11, 477.Google Scholar
- 2.B. Davies and R. G. Storer, Phys. Rev. (1968), 171, 150.Google Scholar
- 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
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