# describe f=u+vi in terms of z

I wish to describe function $f=u+iv$ in terms of complex number z, but this expression seems really hard to convert.
$f=log(x^2+y^2)+(2 \arctan (\frac{y}{x})-2 \arctan (\frac{x}{y})+c)i$, c is a constant.
I know for the first part with log, it can be written to $log(z^2)$ straight away, but how to deal with the second part, it is really hard.