Saddle point Integrals with logarithm and cosine

  • A
  • Thread starter Leo_rr
  • Start date
  • #1
Leo_rr
1
0
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:

Answers and Replies

  • #2
DrDu
Science Advisor
6,258
906
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.
 

Suggested for: Saddle point Integrals with logarithm and cosine

Replies
34
Views
2K
Replies
3
Views
479
Replies
10
Views
585
Replies
1
Views
466
Replies
2
Views
366
Replies
29
Views
585
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
10
Views
1K
  • Last Post
Replies
3
Views
747
Replies
1
Views
419
Top