Let $f \colon \mathbb{C} \longrightarrow \mathbb{C}$ such that $f(2z) = 4\,f(z)$, for all $z \in \mathbb{C}$. Given that $f(0) = 0$ and $f(1) = 1 + 2i$, find $f(\sqrt{2} + i\,\sqrt{2})$.
My attempt was to write $f(z) = \sum a_n\,z^n$ and plug in the conditions, i.e.,
$$\sum a_n\,2^n\,z^n = \sum 4\,a_n\,z^n$$ which gives $a_n\,2^n = 4\,a_n$, but this didn't help at all...
How would I start this question? Any help is appreciated.
$a_n\,2^n = 4\,a_n$ helps a lot, actually. It tells you almost all you need to know. For which $n$ is $a_n\neq0$ at all possible?