Conformal Mapping — Mathematics Knowledge System

Overview

A conformal map is a function that preserves angles locally — if two curves cross at angle θ, their images under the map cross at the same angle θ (including orientation). A key theorem of complex analysis is that any analytic function f with f'(z0) ≠ 0 is automatically conformal at z0: the nonzero derivative acts as a local rotation-and-scaling (multiplication by the complex number f'(z0)), which by definition preserves angles between curves through z0. Conformal maps are central to solving boundary-value problems in physics and engineering, since they can transform a complicated domain (e.g. an airfoil cross-section) into a simple one (e.g. a disk or half-plane) while preserving the harmonic nature of solutions to Laplace's equation.

Intuition

Multiplying a complex number by another complex number w = re^{iθ} rotates it by θ and scales it by r, but crucially never shears it — it treats every direction the same way. Since the derivative f'(z0) of an analytic function is exactly such a single complex multiplier (when nonzero), zooming in close enough to z0, f looks just like multiplication by f'(z0): everything near z0 gets rotated by the same angle and scaled by the same factor, so any angle between two curves crossing at z0 survives unchanged in the image. This is precisely why holomorphy — not mere real differentiability — guarantees angle preservation: a general real map from ℝ² to ℝ² can shear differently in different directions, but an analytic map cannot.

Formal Definition

Definition

A function f: U → ℂ is conformal at z0 ∈ U if it preserves angles (magnitude and orientation) between curves through z0. For analytic f:

f \text{ analytic at } z_0 \text{ and } f'(z_0) \neq 0 \implies f \text{ conformal at } z_0

Analyticity + nonzero derivative implies conformality

f(z_0+h) - f(z_0) \approx f'(z_0)\, h \quad (h \to 0)

Locally, f acts as multiplication by the complex number f'(z0): a rotation by arg(f'(z0)) and scaling by |f'(z0)|

Local linear approximation

Worked Examples

f is entire (analytic everywhere) with derivative f'(z) = 2z.
$f'(z) = 2z$
f'(z) ≠ 0 for all z ≠ 0, so by the conformality criterion, f is conformal at every point except possibly z=0.
$f'(z) \neq 0 \iff z \neq 0$
At z=0, f'(0)=0, so the criterion fails; indeed z² doubles angles at the origin (two curves meeting at angle θ at 0 map to curves meeting at angle 2θ), so conformality breaks down exactly at z=0.
$f'(0) = 0$

Answer: f(z)=z² is conformal at every z ≠ 0; at z=0 the derivative vanishes and angles are doubled rather than preserved.

Practice Problems

Difficulty 5/10

Is f(z) = e^z conformal at z = iπ/2?

Difficulty 6/10

Where does f(z) = sin(z) fail to be conformal?

Difficulty 7/10

Explain, using conformality, why the map w = z² transforms the first-quadrant sector {z : 0 < arg(z) < π/2} onto the upper half-plane.

Quiz

Summary

A conformal map preserves angles (magnitude and orientation) between curves at each point.
Any analytic function with f'(z0) ≠ 0 is automatically conformal at z0, since its local behavior is multiplication by the complex number f'(z0) — a pure rotation and scaling.
Where f'(z0) = 0, conformality typically fails (angles can be multiplied, e.g. doubled for z²), which is why such points are called critical points of the map.

References

WebsiteWikipedia — Conformal map