Complex Analysis – Find Conformal Mapping from Sector to Unit Disc

complex-analysis

Find a conformal mapping between the sector $\{z\in\mathbb{C} : -\pi/4<\arg(z) <\pi/4\}$ and the open unit disc $D$.

I know that it should be a Möbius transformation, but other than that I am very stuck, any help would be much appreciated.

Best Answer

You know that there is a conformal mapping from the unit disk to the upper half plane given by: $$z\mapsto -i\frac{z-i}{z+i}$$ Which sends

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to

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But then you know that the transformation $z\mapsto \sqrt z$ taking the principal value sends the upper half plane to the region you are desiring. This gives:

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Reversing these mappings gives:

$$w \mapsto \frac{iw^2+1}{-w^2-i}$$

Which you will see is a conformal mapping sending the first quadrant to the unit disk.

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