Conformal Mapping — Definition, Formula & Examples

A conformal mapping is a function of a complex variable that preserves angles between curves at every point where it is applied. It stretches or shrinks regions of the complex plane but never distorts the local shape of infinitesimally small figures.

A function $f: U \to \mathbb{C}$ , where $U \subseteq \mathbb{C}$ is open, is conformal at a point $z_0 \in U$ if $f$ is holomorphic (complex-differentiable) at $z_0$ and $f'(z_0) \neq 0$ . At such points, $f$ preserves both the magnitude and the orientation of angles between smooth curves passing through $z_0$ .

Key Formula

w = f(z), \quad f'(z_0) \neq 0

Where:

$z$ = Input point in the complex plane (domain)
$w$ = Output point in the complex plane (image)
$f'(z_0)$ = Derivative of f at z_0; must be nonzero for conformality

How It Works

A conformal map works by transforming one region of the complex plane into another while keeping the angle at which any two curves intersect unchanged. The derivative

f'(z_0)

encodes a local scaling by

|f'(z_0)|

and a rotation by

\arg(f'(z_0))

. Because this scaling and rotation are the same in every direction from

z_0

, the map is locally shape-preserving. To use a conformal map in practice, you choose a known mapping (such as a Möbius transformation or the exponential function) that sends a complicated domain to a simpler one, solve your problem in the simpler domain, and then map the solution back.

Worked Example

Problem: Show that the mapping

f(z) = z^2

is conformal at

z_0 = 1 + i

and find the local scaling factor and rotation angle.

Step 1: Compute the derivative of f.

f'(z) = 2z

Step 2: Evaluate the derivative at

z_0 = 1 + i

and check it is nonzero.

f'(1+i) = 2(1+i) = 2 + 2i \neq 0

Step 3: Find the scaling factor (modulus) and rotation angle (argument).

|f'(1+i)| = |2+2i| = 2\sqrt{2}, \quad \arg(f'(1+i)) = \frac{\pi}{4}

Answer: Since

f'(1+i) \neq 0

, the map

f(z)=z^2

is conformal at

z_0 = 1+i

. It locally scales lengths by a factor of

2\sqrt{2}

and rotates by

\pi/4

radians (45°).

Why It Matters

Conformal mappings are central tools in fluid dynamics, electrostatics, and heat conduction, where Laplace's equation governs the physics. Engineers use them to transform irregular boundaries into simple geometries (like circles or half-planes), solve the problem there, and map the solution back. They also appear in cartography—the Mercator projection is a conformal map of the sphere.

Common Mistakes

Mistake: Assuming

f(z) = z^2

is conformal everywhere, including at the origin.

Correction: At

z = 0

f'(0) = 0

, so the map doubles angles instead of preserving them. Always verify

f'(z_0) \neq 0

at the specific point of interest.