describe f=u+vi in terms of z

Hi, can anyone please help me with this question? Thanks a lot.

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.