Conformal Mappings

A conformal map is a holomorphic function that preserves angles locally. Conformal mappings are fundamental in complex analysis, with applications ranging from fluid dynamics to electrostatics to the uniformization of Riemann surfaces.

Definition and Basic Properties

Definition7.1Conformal map

A holomorphic function $f: D \to \mathbb{C}$ is conformal at $z_0 \in D$ if $f'(z_0) \neq 0$ . At such a point, $f$ preserves the angle between any two smooth curves passing through $z_0$ , both in magnitude and orientation. A map is conformal on $D$ if it is conformal at every point of $D$ .

RemarkAngle preservation

If two curves meet at $z_0$ with tangent directions making angle $\alpha$ , their images under $f$ meet at $f(z_0)$ with the same angle $\alpha$ . This follows because $f$ acts locally as multiplication by $f'(z_0) = |f'(z_0)|e^{i\arg f'(z_0)}$ : a rotation by $\arg f'(z_0)$ and scaling by $|f'(z_0)|$ . Both operations preserve angles.

ExampleExamples of conformal maps

$f(z) = z^2$ is conformal on $\mathbb{C} \setminus \{0\}$ (fails at $z = 0$ where $f'(0) = 0$ and angles are doubled).
$f(z) = e^z$ is conformal on all of $\mathbb{C}$ (since $e^z \neq 0$ ).
The Mobius transformation $f(z) = \frac{az+b}{cz+d}$ ( $ad-bc \neq 0$ ) is conformal on $\mathbb{C} \setminus \{-d/c\}$ .
$f(z) = z + 1/z$ (Joukowski map) is conformal on $\{|z| > 1\}$ and maps the exterior of the unit disk to $\mathbb{C} \setminus [-2, 2]$ .

Mobius Transformations

Definition7.2Mobius transformation

A Mobius transformation (or linear fractional transformation) is a map of the form

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

Mobius transformations form a group under composition isomorphic to $PSL(2, \mathbb{C})$ . They are the automorphisms of the Riemann sphere $\hat{\mathbb{C}}$ .

Theorem7.1Properties of Mobius transformations

Mobius transformations:

Map circles and lines to circles and lines (where lines are "circles through $\infty$ ").
Are uniquely determined by the images of three distinct points.
Preserve the cross-ratio: $(z_1, z_2; z_3, z_4) = \frac{(z_1-z_3)(z_2-z_4)}{(z_1-z_4)(z_2-z_3)}$ .
Are conformal everywhere (on $\hat{\mathbb{C}}$ ).

ExampleMobius transformation from three points

Find a Mobius transformation mapping $1 \mapsto 0$ , $i \mapsto 1$ , $-1 \mapsto \infty$ .

Setting $T(z) = \frac{(z-1)(i+1)}{(z+1)(i-1)}$ : verify $T(1) = 0$ , $T(-1) = \infty$ , $T(i) = \frac{(i-1)(i+1)}{(i+1)(i-1)} = 1$ . So

$T(z) = \frac{(z-1)(1+i)}{(z+1)(i-1)} = \frac{(z-1)(1+i)}{(z+1)(i-1)} \cdot \frac{(-i-1)}{(-i-1)} = \frac{i(z-1)}{z+1}.$

Conformal Maps Between Standard Domains

RemarkImportant conformal maps

Key conformal mappings to know:

Cayley map: $w = \frac{z-i}{z+i}$ maps the upper half-plane $\mathbb{H}$ to the unit disk $\mathbb{D}$ .
Exponential: $w = e^z$ maps the strip $\{0 < \text{Im}(z) < \pi\}$ to the upper half-plane.
Power maps: $w = z^\alpha$ maps a wedge of angle $\pi/\alpha$ to a half-plane.
Joukowski: $w = z + 1/z$ maps $\{|z| > 1\}$ to $\mathbb{C} \setminus [-2,2]$ .
Schwarz-Christoffel: maps the upper half-plane to any polygon.