I got struck on this option of a question in my complex analysis quiz and couldn't correctly solve it.
So, I am asking for help here.
Consider entire function $g(z)=e^z $ $ z\in \mathbb{C}$. Then is $g^{-1}${($z \in \mathbb{C}$; $|z|\leq R$)} bounded for every R>0 or not?
Answer is its not but I am getting that it will be bounded. CAn you please tell how to prove it's not bounded!!
Indeed, the answer is negative. If$$a\in g^{-1}\bigl(\{z\in\Bbb C\mid|z|\leqslant R\}\bigr),$$then$$(\forall n\in\Bbb Z):a+e^{2\pi in}\in g^{-1}\bigl(\{z\in\Bbb C\mid|z|\leqslant R\}\bigr)$$too, since $n\in\Bbb Z\implies e^a=e^{a+2\pi in}$. Therefore, $g^{-1}\bigl(\{z\in\Bbb C\mid|z|\leqslant R\}\bigr)$ is either empty or unbounded. And it is clearly non-empty.