Conformal Mappings

Definition. An analytic function $f$ is conformal at $z_0$ if $f"(z_0) \neq 0$. A conformal Mapping preserves angles (both magnitude and orientation) between curves.

If $f'(z_0) = re^{i\theta}$Then near $z_0$ the mapping $f$ acts as a rotation by $\theta$ followed By a scaling by $r$. The Jacobian determinant is $|f'(z_0)|^2 \gt 0$So orientation is preserved.

A Möbius transformation (or linear fractional transformation) is

$T(z) = \frac{az + b}{cz + d}, \quad ad - bc \neq 0$

Proposition 10.1. Möbius transformations are conformal (where defined) and map circles and lines To circles and lines.

Proposition 10.2. Three points determine a unique Möbius transformation: $T(z_1) = w_1$ $T(z_2) = w_2$, $T(z_3) = w_3$.

Definition. The cross-ratio of four distinct points $z_1, z_2, z_3, z_4$ is

$(z_1, z_2, z_3, z_4) = \frac{(z_1 - z_3)(z_2 - z_4)}{(z_1 - z_4)(z_2 - z_3)}$

Proposition 10.3. The cross-ratio is invariant under Möbius transformations: $(Tz_1, Tz_2, Tz_3, Tz_4) = (z_1, z_2, z_3, z_4)$.

Proposition 10.4. The unique Möbius transformation sending $z_1 \mapsto 0$, $z_2 \mapsto 1$ $z_3 \mapsto \infty$ is

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

A Möbius transformation $T(z) = \frac{az + b}{cz + d}$ is classified by its fixed points (solutions of $T(z) = z$).

1. Parabolic: Exactly one fixed point. Conjugate to $w = z + k$.
2. Elliptic: Two fixed points, $|T'(z_0)| = 1$. Conjugate to a rotation $w = e^{i\theta} z$.
3. Hyperbolic: Two fixed points, $T'(z_0) \in \mathbb{R}^+$, $T'(z_0) \neq 1$. Conjugate to $w = kz$.
4. Loxodromic: Two fixed points, $T'(z_0) \notin \mathbb{R} \cup \{z : |z| = 1\}$. Conjugate to $w = ke^{i\theta}z$.

Theorem 10.5 (Riemann Mapping Theorem). Let $U$ be a connected open proper subset of $\mathbb{C}$. Then there exists a bijective conformal map from $U$ onto the unit disk $\mathbb{D} = \{z : |z| \lt 1\}$.

This is one of the most profound results in complex analysis, establishing that all connected Domains (other than $\mathbb{C}$ itself) are conformally equivalent.

Remark. The Riemann mapping theorem is an existence theorem; it does not provide an explicit Formula for the conformal map .

Fluid dynamics. The complex potential $w = f(z) = \phi + i\psi$ for a 2D incompressible, irrotational flow satisfies Laplace’s equation. Conformal mappings transform simple flow patterns (e.g., uniform flow past a circle) into flows past arbitrary smooth boundaries. The Joukowski transform $w = z + 1/z$ maps a circle to an airfoil shape, enabling analytical calculation of lift.

Electrostatics. The electric potential in a charge-free region satisfies $\nabla^2 V = 0$. Conformal mappings transform the boundary value problem into a simpler geometry (e.g., upper half-plane or unit disk) where the solution is known, then map the solution back.

Heat conduction. Steady-state temperature distributions satisfy Laplace’s equation. Conformal mappings solve heat flow problems in irregularly shaped regions by mapping to canonical domains.

Key insight: Any problem governed by Laplace’s equation in 2D can be solved by conformally mapping the domain to a half-plane or disk, solving there, and mapping back.

Problem. Use the Joukowski transform to find the complex potential for flow past a cylinder.

Solution. The complex potential for uniform flow past a circle of radius $R$ centered at the origin is:

$w(\zeta) = U\left(\zeta + \frac{R^2}{\zeta}\right)$

The Joukowski transform $z = \zeta + \frac{1}{\zeta}$ maps the circle $|\zeta| = R$ to an ellipse (or airfoil for $R$ near 1 with slight offset). Substituting $\zeta$ as a function of $z$ and composing gives the flow past the transformed body.

The velocity components are obtained from:

$v_x - iv_y = \frac{dw}{dz} = \frac{dw}{d\zeta}\cdot\frac{d\zeta}{dz}$

At infinity, $v_x = U$ and $v_y = 0$ (uniform flow). On the cylinder surface, the flow is tangent to the boundary (no penetration condition). $\blacksquare$