TITLE: What Möbius transformation maps the unit circle $\{z: |z|=1\}$ to the real axis?
QUESTION [10 upvotes]: Can you determine a Möbius transformation that maps unit circle $\{z: |z|=1\} \rightarrow$ real axis. 
I.e., how would you find one?
Would this transformation be uniquely determined? 
The Möbius transformations are the maps of the form: $$ f(z)= \frac{az+b}{cz+d}.$$

REPLY [11 votes]: Try sending $1$ to $0$ and $-1$ to $\infty$:
$$
\frac{z-1}{z+1}\tag{1}
$$
Compute the real part of $(1)$:
$$
\begin{align}
\frac12\left(\frac{z-1}{z+1}+\frac{\bar{z}-1}{\bar{z}+1}\right)
&=\frac12\frac{(z\bar{z}+(z-\bar{z})-1)+(z\bar{z}-(z-\bar{z})-1)}{|z+1|^2}\\
&=\frac{|z|^2-1}{|z+1|^2}\tag{2}
\end{align}
$$
Thus, if $|z|=1$, $(2)$ says that $\mathrm{Re}\left(\frac{z-1}{z+1}\right)=0$; that is, $(1)$ sends the unit circle to the imaginary axis. Therefore,
$$
i\frac{z-1}{z+1}\tag{3}
$$
sends the unit circle to the real axis.