Complex Analysis or Complex Methods | Part IB, 2005

Determine a conformal mapping from Ω0=C\[1,1]\Omega_{0}=\mathbf{C} \backslash[-1,1] to the complex unit disc Ω1={zC:z<1}.\Omega_{1}=\{z \in \mathbf{C}:|z|<1\} .

[Hint: A standard method is first to map Ω0\Omega_{0} to C\(,0]\mathbf{C} \backslash(-\infty, 0], then to the complex right half-plane {zC:Rez>0}\{z \in \mathbf{C}: \operatorname{Re} z>0\} and, finally, to Ω1.]\left.\Omega_{1} .\right]

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