Skip to content

Conformal Mapping

Source lecture(s): PHY622 Lec4

Intuition

An analytic function with non-zero derivative warps the plane while preserving angles—turning hard domains into easy ones.

Formal Definition

A mapping \(w=f(z)\) is conformal if it preserves angles between intersecting curves.

Mathematical Formulation

Conformal iff \(f'(z)\neq0\) everywhere. Example: \(w=z^2\) maps the upper half-plane to the plane with a branch cut.

Derivation

Near \(z_0\), \(f(z)\approx f(z_0)+f'(z_0)(z-z_0)\). Multiplying by \(f'(z_0)\) rotates and scales but preserves angles.

Worked Example

The Joukowski map \(w=z+1/z\) transforms a circle to an airfoil shape.

Common Mistakes

  • Forgetting that conformality fails where \(f'(z)=0\).
  • Applying to non-analytic transformations.

Quiz

Q1. What breaks conformality?

Answer

Points where \(f'(z)=0\).