# How to calculate a complex integral with branch cut?

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.

• Please consider clarifying your question. What are the the variables $x$, $x'$, $y$, $y'$? To render LaTeX, please see here. – Mårten W Oct 17 '14 at 9:10
• I hope my edit is correct, please edit it if it is wrong. – BlackAdder Oct 17 '14 at 9:14
• x , x' ,y, y' are cartesian coordinate's points. Firstly we can think x-x' = a (constant) and y-y' = b (constant). – hazara Oct 17 '14 at 9:54

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