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A Saddle point Integrals with logarithm and cosine

  1. Jun 13, 2016 #1
    I'm trying to use the saddle point method to solve the following integral:

    Z = (1/sqrt{2 pi t}) ∫_{1}^{infinity} ds (1/sqrt{2 pi s}) exp{ p [-s ln(s/t) +s] } cos(2 pi L~ p ~ s), as p → infinity
    Mod edit to make integral more readable:
    $$Z = \frac{1}{\sqrt{2\pi t}} \int_1^{\infty} \frac 1 {\sqrt{2 \pi s}} e^{p(-s \ln(s/t) + s)} \cos(2 \pi L p s)ds$$
    $$\lim_{p \to \infty}Z = ? $$
    Is this the correct integral?
    where L and p are Integers. If we remove the fast oscillatory cosine, the Sadde Point method is straightforward and the solution is:

    Z = e^{pt}

    Introducing a fast oscillatory function in the integral, one would expect the value of Z to be much smaller, but the direct (or maybe naive) application of the method results in the exact same result, and that's absurd! Have anybody faced a similar problem?
    Last edited by a moderator: Jun 13, 2016
  2. jcsd
  3. Jun 17, 2016 #2


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    Science Advisor

    I think you should write the cosine as the sum of two exponentials which can then be combined with the other exponential and you get two integrals of which each can be evaluated by the saddle point approximation.
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