# Friday April 3rd

## Conformal Mappings

> Qual Alert: basically everything today.
> E.g. Show that the conformal maps $\CC\to \CC$ are linear.

The three basic types are translation, dilations, and rotation.

For any $n\in \NN$, the map $z\mapsto z^n$ is conformal in the sector $S = \theset{z \suchthat 0 < \arg(z) < {\pi \over n}}$.
For $0 < \alpha < 2$, the map takes $\HH$ to $\theset{0 < \arg{z} < \alpha\pi}$.

The map $z \mapsto {1 +z \over 1 - z}$ takes the upper half-disc $\DD^+$ to the first quadrant $\HH^+$.
The inverse is $w- 1 \over w +1$ which is clearly holomorphic for $w\neq 1$, and its image is contained in the unit disc since the distance from $w$ to $-1$ is always greater than the distance from $w$ to $1$.

Note that
\begin{align*}
f\left(e^{i \theta}\right)=\frac{1+e^{i \theta}}{1-e^{i \theta}}=\frac{e^{-i \theta / 2}+e^{i \theta / 2}}{e^{-i \theta / 2}-e^{i \theta / 2}}=\frac{2 \cos \frac{\theta}{2}}{-2 i \sin \frac{\theta}{2}}=i \cot \frac{\theta}{2}
\end{align*}

and thus as $\theta$ travels from $0$ to $\pi$ along $S^1$, $f(e^{i\theta})$ travels from $i\infty$ to 0.

The logarithm $z\mapsto \log z$ (taking the branch cut along the negative imaginary axis) maps $\HH$ to the strip $\theset{z \suchthat x\in \RR, 0 < y < \pi}$.
The inverse is given by $w \mapsto e^w$, and as $x$ travels from $-\infty$ to $0$, $f(x)$ travels from $\infty + i\pi$ to $-\infty + i\pi$.
When $x$ travels from $0\to\infty$, $f(x)$ travels from $-\infty_\RR \to \infty_\RR$.


The map $z \mapsto \log z$ also defined as map from the half disc $\DD^+$ to the half-strip $\SS = \theset{x <0, 0 < y < \pi}$.
We have $[1, -1]_\RR \mapsto [-\infty ,0]$ and $S^1_+ \mapsto i[0, \pi]$.

The map $z \mapsto -{1 \over 2}\qty{z + {1 \over z}}$ takes $\DD^+ \to \HH$.
We have $[0, 1] \mapsto [1 ,\infty]$ and $S^1_+ \mapsto [-1, 1]\subset \RR$, and $[0, -1] \mapsto [-1, -\infty] \subset \RR$.


The map $z\mapsto \sin z$ takes $\theset{-\pi/2 < x < \pi/2, y>0}$ onto $\HH$.
Then $[-\pi/2 + i\infty, -\pi/2 + 0] \mapsto [-\infty, 1]$ and $[-\pi/2, \pi/2] \mapsto [-1, 1]$, and $[\pi/2 + 0i, \pi/2 + o\infty] \mapsto [1, \infty]$.


> Just need to know these 8 examples of conformal mappings.