Conformal Mappings

10.1 Definition

Definition. An analytic function $f$ is conformal at $z_0$ if $f"(z_0) \neq 0$ . A conformal Mapping preserves angles (both magnitude and orientation) between curves.

10.2 Geometric Interpretation

If $f'(z_0) = re^{i\theta}$ Then near $z_0$ the mapping $f$ acts as a rotation by $\theta$ followed By a scaling by $r$ . The Jacobian determinant is $|f'(z_0)|^2 \gt 0$ So orientation is preserved.

10.3 Common Conformal Mappings

Mapping	Effect
$w = az + b$ ( $a \neq 0$ )	Translation, rotation, scaling
$w = 1/z$	Inversion in the unit circle
$w = z^2$	Squaring (doubles angles)
$w = e^z$	Exponential (maps strips to sectors)
$w = \frac{z - a}{1 - \bar{a}z}$	Möbius (maps disk to disk)

10.4 Möbius Transformations

A Möbius transformation (or linear fractional transformation) is

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

Proposition 10.1. Möbius transformations are conformal (where defined) and map circles and lines To circles and lines.

Proposition 10.2. Three points determine a unique Möbius transformation: $T(z_1) = w_1$ $T(z_2) = w_2$ , $T(z_3) = w_3$ .

10.5 Cross-Ratio

Definition. The cross-ratio of four distinct points $z_1, z_2, z_3, z_4$ is

$(z_1, z_2, z_3, z_4) = \frac{(z_1 - z_3)(z_2 - z_4)}{(z_1 - z_4)(z_2 - z_3)}$

Proposition 10.3. The cross-ratio is invariant under Möbius transformations: $(Tz_1, Tz_2, Tz_3, Tz_4) = (z_1, z_2, z_3, z_4)$ .

Proposition 10.4. The unique Möbius transformation sending $z_1 \mapsto 0$ , $z_2 \mapsto 1$ $z_3 \mapsto \infty$ is

$T(z) = \frac{(z - z_1)(z_2 - z_3)}{(z - z_3)(z_2 - z_1)}$

10.6 Classification of Möbius Transformations

A Möbius transformation $T(z) = \frac{az + b}{cz + d}$ is classified by its fixed points (solutions of $T(z) = z$ ).

Parabolic: Exactly one fixed point. Conjugate to $w = z + k$ .
Elliptic: Two fixed points, $|T'(z_0)| = 1$ . Conjugate to a rotation $w = e^{i\theta} z$ .
Hyperbolic: Two fixed points, $T'(z_0) \in \mathbb{R}^+$ , $T'(z_0) \neq 1$ . Conjugate to $w = kz$ .
Loxodromic: Two fixed points, $T'(z_0) \notin \mathbb{R} \cup \{z : |z| = 1\}$ . Conjugate to $w = ke^{i\theta}z$ .

Solution

Problem. Find the Möbius transformation mapping $0 \mapsto i$ , $1 \mapsto 0$ , $\infty \mapsto -i$ .

$T(z) = \frac{az + b}{cz + d}$ with $T(0) = i \Rightarrow b/d = i \Rightarrow b = id$ . $T(1) = 0 \Rightarrow a = -b = -id$ . $T(\infty) = -i \Rightarrow a/c = -i \Rightarrow c = d$ .

$T(z) = \frac{-idz + id}{dz + d} = \frac{i(1 - z)}{z + 1}$ .

Problem. Show that $T(z) = \frac{z - 1}{z + 1}$ maps the right half-plane to the unit disk.

If $\mathrm{Re}(z) \gt 0$ Then $|z - 1| \lt |z + 1|$ So $|T(z)| \lt 1$ .

Check boundary: $T(i) = \frac{i - 1}{i + 1} = \frac{(i-1)(-i+1)}{(i+1)(-i+1)} = \frac{2}{2} = 1$ . $|T(i)| = 1$ . $\checkmark$

Problem. Classify $T(z) = \frac{2z + 1}{z + 2}$ .

Fixed points: $z = \frac{2z + 1}{z + 2} \Rightarrow z^2 = 1 \Rightarrow z = \pm 1$ .

$T'(z) = \frac{3}{(z + 2)^2}$ . $T'(1) = 1/3$ , $T'(-1) = 3$ .

Both multipliers are real and positive (not equal to $1$ ), so $T$ is hyperbolic.

10.7 The Riemann Mapping Theorem

Theorem 10.5 (Riemann Mapping Theorem). Let $U$ be a connected open proper subset of $\mathbb{C}$ . Then there exists a bijective conformal map from $U$ onto the unit disk $\mathbb{D} = \{z : |z| \lt 1\}$ .

This is one of the most profound results in complex analysis, establishing that all connected Domains (other than $\mathbb{C}$ itself) are conformally equivalent.

Remark. The Riemann mapping theorem is an existence theorem; it does not provide an explicit Formula for the conformal map .

Mark this page as reviewed