TITLE: Is it a harmonic function or not?
QUESTION [6 upvotes]: I am trying to resolve a question whether a certain function is harmonic or not. If yes, I should find its harmonic conjugate.
The function is $u = \frac{x}{x^2+y^2}$.
I found that it is a harmonic function by using Laplace equation, but I am not sure. How can I find its harmonic conjugate, please help anyone.

REPLY [3 votes]: You can do this without calculation. Recall that a holomorphic function is harmonic.
The function
$$
f:z\longmapsto \frac{1}{z}
$$
is holomorphic, so it is harmonic.
Hence its real part
$$
u:z=x+iy\longmapsto \mbox{Re}f(z)=\mbox{Re}\frac{\bar{z}}{|z|^2}=\frac{\mbox{Re} \bar{z}}{|z|^2}=\frac{x}{x^2+y^2}
$$
is harmonic.
And the harmonic conjugate of $u$ is
$$
v:z=x+iy\longmapsto \mbox{Im} f(z) =\frac{\mbox{Im} \bar{z}}{|z|^2}=-\frac{y}{x^2+y^2}.
$$