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