TITLE: How to calculate this integral
QUESTION [5 upvotes]: I have a function $u(x,y)$ defined on the domain $|x|<\infty, y>0$. I know that
$$ \frac{\partial u(x,y)}{\partial y} = \frac{y}{\pi}\int_{-\infty}^{\infty} \frac{f(w)}{y^2 + (x-w)^2}dw$$
How can I calculate $u(x,y)$? I have no idea how to tackle this integral.

REPLY [2 votes]: You haven't said anything about the function $f(w)$ (you really should).
If $f$ is holomorphic in the complex upper half-plane, and is bounded at infinity, $f(w)=O(1)$, $|w|\to\infty$, $\Im w\geq0$, then you can use Cauchy theorem to write
$$ \frac{y}{\pi} \int_{-\infty}^\infty \frac{f(w)}{(w-x)^2+y^2}\,dw
= 2iy\,\text{Res}_{w=x+iy}\frac{f(w)}{(w-x)^2+y^2}
= f(x+iy),$$
so that your function $u(x,y)$ can be written as
$$ u(x,y) = g(x) + \int^y f(x+iy')\,dy', $$
where $g(x)$ is an arbitrary function of $x$.
Other things will also happen depending on the properties of $f(w)$.