An integration in PDE

Question

$$\dfrac{y}{\pi}\int_{-\infty}^{+\infty}\dfrac{{\rm d}\xi}{(1+\xi^2)\left[(\xi-x)^2+y^2\right]}$$

I am stuck here. Any help would be appreciated!

Answer 1

Outline of the solution:

Suppose $x$ and $y$ are real numbers and $(x,y)\ne(0,1)$ and denote the integral by $I$. Take the upper hemispherical contour in the complex plane. The part at infinity vanishes. WLOG $y>0$. Let the integrand be $f(z)$. \begin{align}\frac{\pi}{y}2\pi iI &= \sum\text{Res}[f(\text{poles above the real axis})] \\ &= \frac{1}{2i((i-x)^2+y^2)}+\frac1{(1+(x+iy)^2)(2iy)}. \end{align}