Compute $f''(0)$ for a holomorphic function on a square given $f'(0)$ and $f(0)$

Question

Let $S$ be the square $\{x + iy: |x| < 1, |y| < 1\}$ and $f:S \rightarrow S$ a holomorphic function so that $f(0)= 0$ and $f'(0) = 1$. Find $f''(0)$.

It seems like I need to use Cauchy's integral formula but I'm not sure. I guess I'm missing what information I gain by having $f(0)$ and $f'(0)$.

Answer 1

The square is a red herring; it could be any domain whatsoever. The key point is the uniqueness part of the Riemann mapping theorem (as Lukas Geyer hinted):

For any simply connected domain $\Omega$ (other than $\mathbb C$) and a point $a\in \Omega$ there is a unique conformal map $f$ from $\Omega$ onto the unit disk such that $f(a)=0$ and $f'(a)>0$.

Considering the appropriate composition of maps and using the above statement, you should be able to show the following:

For any two simply connected domains $\Omega_1$, $\Omega_2$ (other than $\mathbb C$) and points $a_k\in \Omega_k$, $k=1,2$ there is a unique conformal map $f$ from $\Omega_1$ onto $\Omega_2$ such that $f(a_1)=a_2$ and $f'(a_1)>0$.

How does this help finding $f''(a_1)$? In general it doesn't. But in your case, $\Omega_1=\Omega_2$ and $a_1=a_2$, so you have one map with the above properties: the identity map. The uniqueness statement tells you that this is $f$.