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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 $arg f'(z_0)=0$


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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 are a good start. – leslie townes Jan 15 '12 at 0:56

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