I have a Dirichlet problem on a half-plane
$\begin{equation*} \begin{cases} \Delta{u}=0, y>0 \\ u(x,0) = \frac{x}{1+x^2}, y =0 \end{cases} \end{equation*}$
And in solving this PDE I have integral and answer, but the author doesn't show how he solved this problem
$u(x_0, y_0, z_0)=-\frac{y}{\pi}\int_{-\infty}^{+\infty} \frac{x}{1+x^2} \frac{1}{(x-x_0)^2+y^2_0} dx=\frac{x_0}{x_0^2 + (1+y_0)^2}$
I want to take the integral but I don't know what $x_0 \; and\; y_0$ means when I take the integral. The author wrote that solved by "The theory of residue integrals". But I don't know why he uses it here because here we don't have complex numbers.