# showing that there exist only one function $f$

need some help with this question: Let $\Omega \neq \mathbb C$ a simply connected area. Let $w_0 \in \Omega, z_0 \in D=\{z: |z|<1\}$ and $-\pi<\theta<\pi$

I want to show that there exist only one function $f$ such that: $f$ is bijection, $f(D)=\Omega$, $f(z_0)=w_0$ and $arg f'(z_0)=0$

Thanks.

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What is $\theta$? Where are you stuck: existence or uniqueness? –  Davide Giraudo Jan 14 '12 at 21:22
Did you mean $\text{arg} f'(z_0) = \theta$ ? –  Joel Cohen Jan 14 '12 at 22:39
Presumably you want $f$ to be holomorphic on all of $D$ (not just complex differentiable at $z_0$, which is all that can be inferred from what you wrote). Without this condition there is no uniqueness (and no conceivable use of the result, really). I suggest web and textbook searches for "Riemann mapping theorem". The references in en.wikipedia.org/wiki/Riemann_mapping_theorem are a good start. –  leslie townes Jan 15 '12 at 0:56