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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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    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 http://en.wikipedia.org/wiki/Riemann_mapping_theorem are a good start.2012-01-15

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