How to calculate a complex integral with branch cut?

Question

I want to calculate this integral:

$$\int_{-\infty}^{\infty} \frac{\exp\left(-i\sqrt{(\xi^2 - k^2)(y -y')} -i\xi(x - x')\right)}{-i\sqrt{\xi^2 - k^2}}d\xi$$

I can not calculate this integral, I can try to calculate with Matlab, but I don't know exactly Matlab code. Does anyone have any advice for me? We can think x-x' and y-y' are constant.

Answer 1

1. First fold the integral function about zero by substituting $\xi = -\xi$ for $\xi < 0$.
2. Transform the integral contour from $0 \to \infty$ to $0 \to 1$ using a transform function like $\frac{s^2}{1 - s^2}$.
3. Add small imaginary component to $k$ to account for the Cauchy Principle value. It is equivalent to calculate the principle value integral.
4. Use quadgk or quadl, whatever is suitable to integrate exclusing the end points where the function is not defined.
5. Please download the whole folder 'matlab' which contains the main code 'main.m' and several other routines from the link http://www.win.tue.nl/~dsingh/images/matlab/

The attached code will do your calculations for various values of $k$.

Ps: If you find it difficult to understand the code, let me know.