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Suppose I have a sector $D = \{0 < \arg z < \alpha\}$ where $\alpha \leq 2\pi$. If I apply the function $w = \frac{\zeta - i}{\zeta + i}$ from the upper half plane to the unit disc ($\zeta = z^{\frac{\pi}{\alpha}}$), I get that the vertex of the sector goes to -1 and $z = \infty$ goes to 1. I get the unit circle essentially. My question for this example is: How do we know that the angles are preserved?

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