Conformal Mappings
Conformal Mappings
An analytic function $f$ with $f'(z)\neq 0$ is a conformal map: it preserves angles between curves. Conformal maps are used to transform difficult boundary value problems (e.g., Laplace's equation on complicated domains) into problems on simple domains like the unit disk or upper half-plane. The Riemann mapping theorem guarantees that every simply connected proper subset of $\mathbb{C}$ is conformally equivalent to the unit disk.
Conformal Mapping
A function $f:D\to\mathbb{C}$ is conformal at $z_0$ if $f'(z_0)\neq 0$ — it maps infinitesimal disks to infinitesimal disks (scaled by $|f'|$ and rotated by $\arg f'$), preserving angles. The Möbius transformation $f(z)=\frac{az+b}{cz+d}$ (with $ad-bc\neq 0$) maps circles and lines to circles and lines, and any three points can be mapped to any three other points. The Joukowski map $z\mapsto z+1/z$ maps the unit circle to the interval $[-2,2]$ and near-circular contours to airfoil shapes (aerodynamics).
Riemann Mapping Theorem
Every simply connected proper open subset $U\subsetneq\mathbb{C}$ is conformally equivalent to the open unit disk $\mathbb{D}=\{z:|z|<1\}$: there exists a bijective analytic $f:U\to\mathbb{D}$ whose inverse is also analytic (biholomorphism). The map is unique once normalized (e.g., $f(z_0)=0$, $f'(z_0)>0$ for a given $z_0\in U$). This theorem reduces Dirichlet problems on arbitrary simply connected domains to those on $\mathbb{D}$, which can be solved by the Poisson integral formula.
Example 1
Find the Möbius transformation mapping $z=0\to w=i$, $z=1\to w=0$, $z=-1\to w=\infty$.
Solution: General Möbius: $w=\frac{az+b}{cz+d}$. $z=-1\to\infty$ means $c(-1)+d=0$, so $d=c$. Then $w=\frac{az+b}{c(z+1)}$. $z=1\to 0$: $a+b=0$, so $b=-a$: $w=\frac{a(z-1)}{c(z+1)}$. $z=0\to i$: $\frac{a(-1)}{c}=i$, so $a/c=-i$. Set $c=1$: $w=\frac{-i(z-1)}{z+1}=\frac{i(1-z)}{1+z}$.
Example 2
Show the map $w=e^z$ maps the horizontal strip $\{z:0<\text{Im}(z)<\pi\}$ to the upper half-plane.
Solution: $z=x+iy$, $y\in(0,\pi)$. $w=e^x e^{iy}$: modulus $e^x>0$ (all positive reals), argument $y\in(0,\pi)$. So $\text{Im}(w)=e^x\sin y>0$ (since $\sin y>0$ for $y\in(0,\pi)$). As $x$ ranges over $\mathbb{R}$, $e^x$ ranges over $(0,\infty)$. So the image is $\{w:|w|>0,0<\arg w<\pi\}=$ upper half-plane $\{\text{Im}(w)>0\}$.
Practice
- Find the Möbius transformation mapping the unit disk $|z|<1$ to the upper half-plane.
- Show that Möbius transformations form a group under composition.
- Apply the Joukowski map to determine the lift on a circular cylinder in potential flow.
- Use conformal mapping to solve $\Delta u=0$ on the quarter-plane $\{x>0,y>0\}$ with $u=0$ on $\{y=0\}$ and $u=1$ on $\{x=0\}$.
Show Answer Key
1. $w=\frac{z-i}{z+i}$ maps $|z|=1$ to the real line and $|z|<1$ to $\text{Im}(w)>0$ (the upper half-plane). Its inverse is $z=i\frac{1+w}{1-w}$.
2. Composition: if $T_1(z)=\frac{az+b}{cz+d}$ and $T_2(z)=\frac{\alpha z+\beta}{\gamma z+\delta}$, then $T_1\circ T_2$ has the same form (matrix multiplication of $\begin{pmatrix}a&b\\c&d\end{pmatrix}\begin{pmatrix}\alpha&\beta\\\gamma&\delta\end{pmatrix}$). Identity is $z$, inverse is $\frac{dz-b}{-cz+a}$. Associativity follows from matrix multiplication.
3. The Joukowski map $w=z+1/z$ maps a circle $|z|=R>1$ to an ellipse and $|z|=1$ to the flat plate $[-2,2]$. By conformal mapping, the flow around the cylinder maps to flow around an airfoil, and lift is computed via the Kutta-Joukowski theorem: $L=\rho U\Gamma$ per unit span.
4. Map the quarter-plane to UHP via $w=z^2$, then solve $\Delta u=0$ on UHP with $u=0$ on positive real axis and $u=1$ on negative real axis. Solution: $u=\frac{1}{\pi}\arg(w)=\frac{2}{\pi}\arg(z)=\frac{2}{\pi}\arctan(y/x)$.