I'm trying to solve the following exercise from Rudin's Real & Complex Analysis, chapter 14 exercise 22.
Suppose $f$ is a one-to-one conformal mapping of $U$ onto a square with center at $0$, and $f(0) = 0$. Prove that $f(iz) = if(z)$. If $f(z) = \sum c_n z^n$, prove that $c_n = 0$ unless $n – 1$ is a multiple of $4$. Generalize this: Replace the square by other simply connected regions with rotational symmetry.
The part about $c_n = 0$ is easy once the first part is proved. This is were I'm stuck. I suspect the generalization will become easier once I know how to prove the first part. Any help is appreciated.
Notes:
- This is not homework. I'm reading the text on my own.
- $U$ is the unit disc.
Best Answer
Hint: $g(z)=f^{-1}(if(z))$ is a conformal 1-1 map from $U$ onto itself fixing the origin. Try playing a little with this or something similar.