Conformal Mappings

Introduction

A conformal mapping preserves angles. When two curves cross at 45°, their images under a conformal map also cross at 45°. This angle-preserving property makes conformal mappings invaluable for solving problems in physics—particularly in fluid dynamics, electrostatics, and heat flow.

In complex analysis, conformal mappings are precisely the analytic functions with nonzero derivative. The Riemann mapping theorem guarantees that any simply connected domain (except the whole plane) can be conformally mapped to the unit disk.

Definition

A function ƒ is conformal at (z_0) if it preserves angles. In complex analysis, this requires ƒ to be analytic with ƒ′*((z_0))≠0.

ƒ(z)≈ƒ((z_0))+ƒ′*((z_0))*(z−(z_0))

Multiplication by ƒ′*((z_0))=r*e(i*θ) rotates by θ and scales by r—both preserve angles.

Important Examples

Linear Functions

ƒ(z)=a*z+b*(a≠0)

Rotation by arg(a), scaling by |a|, translation by b. Maps circles to circles, lines to lines.

Power Functions

ƒ(z)=zn

Multiplies angles by n. A wedge of angle π/n maps to a half-plane. Conformal except at z=0.

Exponential Function

ƒ(z)=ez

Maps horizontal strips of height 2*π to the punctured plane. The strip 0<Im(z)<π maps to the upper half-plane.

Möbius Transformations

ƒ(z)=(a*z+b)/(c*z+d)

Map circles to circles. The Cayley transform w=(z−i)/(z+i) maps the upper half-plane to the unit disk.

Joukowski Transform

ƒ(z)=z+1/z

Famous for generating airfoil shapes in aerodynamics. Maps circles to ellipses and airfoil profiles.

The Riemann Mapping Theorem

Any simply connected domain D⊂C (other than C itself) can be conformally mapped onto the unit disk. This is one of the most remarkable theorems in complex analysis.

The mapping is unique if we specify: ƒ((z_0))=0 and ƒ′*((z_0))>0 for some (z_0)∈D.

The theorem is existential—it guarantees a conformal map exists but does not provide an explicit formula. Finding explicit mappings for specific domains is often challenging.

The Schwarz-Christoffel Formula

Maps the upper half-plane to a polygon. For a polygon with interior angles (α_1)*π,(α_2)*π,…,(α_n)*π at vertices (w_1),…,(w_n):

ƒ(z)=A*(∫_^)((z−(x_1))((α_1)−1)*(z−(x_2))((α_2)−1)⋯d(z))+B

where (x_1),(x_2),… are prevertices on the real axis.

Applications

Fluid Dynamics

For 2D ideal fluid flow, the complex potential satisfies Laplace's equation. Conformal maps transform flow around complicated obstacles into flow around simple shapes.

Electrostatics

Electric potential also satisfies Laplace's equation. Conformal maps transform boundary value problems on complex domains to the disk or half-plane where solutions are known.

Heat Conduction

Steady-state temperature distributions also satisfy Laplace's equation, making conformal mapping a powerful tool for heat transfer problems.

Properties Preserved

• Angles (by definition)

• Harmonic functions (solutions to Laplace's equation)

• Local shape (infinitesimally)

• NOT preserved: distances, areas, straight lines (in general)

Summary

Conformal mappings are analytic functions with nonzero derivative. They preserve angles and transform harmonic functions, making them essential for solving Laplace's equation on complex domains.

Key examples include linear maps, powers, exponentials, Möbius transformations, and the Joukowski transform. The Riemann mapping theorem guarantees existence of conformal maps to the disk, while Schwarz-Christoffel provides explicit formulas for polygonal domains.