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 \(\mathbb{D}=B_1(0)\).