TITLE: Is $\frac{z-\alpha}{1-\overline{\alpha}z}$ some special function in complex analysis?
QUESTION [5 upvotes]: Many homework problems seem to use the following function (or something very close to it):
$$F_\alpha(z)=\frac{z-\alpha}{1-\overline{\alpha}z}$$
Does it serve some special purposes in complex analysis?
It does revolve around the unit circle.

REPLY [3 votes]: This is "the" conformal map from $\mathbb{D}$ to itself that sends $a$ with $0$, and is unique up to rotation. Applications include things such as the Schwarz-Pick theorem.