# Asymptotic approximation to a closed contour integral

1. Oct 17, 2008

### bdforbes

Find an asymptotic approximation as p goes to infinity:

$$f_{\lambda}(p)=\oint_{C}exp(-ipsinz+i\lambda z)dz$$

where C is a square contour and p, lambda are real.

Taking C to be of side length pi and centered at the origin, I applied the method of steepest descent at the point z=-pi/2, since as p goes to infinity, the contributions from all other points on the contour become negligible. I got the answer given by the lecturer, but I don't know how to interpret this.

This is a closed contour integral around a compact domain, of an analytic function. It is equal to zero for all finite values of p, by Cauchy's Theorem. However, the asymptotic solution is claiming that for large p, the value of the integral is going roughly like:

$$p^{-\frac{1}{2}}e^{ip}$$

This goes to zero as p goes to infinity, but why should the integral ever be non-zero?