# Conformal mappings

Personal notes on conformal mappings in complex analysis.

Notes taken while taking a course [1].

## Conformal mapping

A smooth complex function $$f$$ is conformal at $$z$$ if for any two curves $$\gamma_1$$ and $$\gamma_2$$ that intersect at $$z$$ with non-zero tangents, then the tangents of $$f \circ \gamma_1$$ and $$f \circ \gamma_2$$ intersects at $$f(z)$$ with the same angle.

A conformal mapping from $$D$$ to $$V$$ is a bijection which is conformal on all points in $$D$$.

If $$f : D \rightarrow \Complex$$ is analytic and $$f'(z_0) \neq 0$$ for $$z_0 \in D$$, then $$f$$ is conformal at $$z_0$$.

## Möbius transformations

A Möbius transformation is a function $$f$$ of the form

$f(z) = \frac{az + b}{cz + d}$

where $$\{a,b,c,d\} \subset \Complex$$ and $$ad-bc \neq 0$$. A Möbius transformation is also called a fractional linear transformation.

Möbius transformation are conformal mappings from $$\hat{\Complex}$$ to $$\hat{\Complex}$$, and in fact they are the only conformal mappings from $$\hat{\Complex}$$ to $$\hat{\Complex}$$.

### Extended complex plane

Includes $$\infty$$ in addition to the complex numbers. In other words,

$\hat{\Complex} = \Complex \cup \{ \infty \}$

### Affine transformations

An affine transformation is a Möbius transformation $$f$$ of the form

$f(z) = az + b$

where $$a \neq 0$$.

Affine transformations are conformal mappings from $$\Complex$$ to $$\Complex$$, and are in fact the only conformal mappings from $$\Complex$$ to $$\Complex$$.

### Mapping distinct points

For distinct points $$z_1$$, $$z_2$$, $$z_3$$, there is a unique Möbius transformation $$f$$ where

$f(z) = \frac{z-z_1}{z-z_3} \cdot \frac{z_2 - z_3}{z_2 - z_1}$

which maps $$z_1$$, $$z_2$$, $$z_3$$ to $$0$$, $$1$$, $$\infty$$, respectively.

For distinct points $$z_1$$, $$z_2$$, $$z_3$$ and distinct points $$w_1$$, $$w_2$$, $$w_3$$, there is a unique Möbius transformation which maps $$z_1$$, $$z_2$$, $$z_3$$ to $$w_1$$, $$w_2$$, $$w_3$$, respectively.

### Composition

The composition of two Möbius transformations is also a Möbius transformation.

Any Möbius transformation can be composed from the following three types of transformations:

\begin{align*} f(z) &= az && \text{(rotation and dilation)} \\ f(z) &= z+b && \text{(translation)} \\ f(z) &= \frac{1}{z} && \text{(inversion)} \end{align*}

### Mapping shapes

Möbius transformations map circles and lines to circles and lines.

## Riemann mapping theorem

If $$D$$ is a simply connected domain in the complex plane, i.e., open, connected, and has no holes, and $$D$$ is a strict subset of $$\Complex$$, then there is a conformal mapping from $$D$$ onto the unit disk $$\mathbb{D}=B_1(0)$$.

## References

[1]
Petra Bonfert-Taylor. Introduction to complex analysis. Course on Coursera.