Multiplicative but non-additive function $f : \mathbb{C} \to \mathbb{R}$ I'm trying to find a function $f : \mathbb{C} \to \mathbb{R}$ such that 


*

*$f(az)=af(z)$ for any $a\in\mathbb{R}$, $z\in\mathbb{C}$, but

*$f(z_1+z_2) \ne f(z_1)+f(z_2)$ for some $z_1,z_2\in\mathbb{C}$.


Any hints or heuristics for finding such a function?

HINT: Look at $z$ in polar form.
