Conformal Mappings

Motivation: What Does a "Correct" Mapping Mean

Many problems in mathematical physics are posed in complex domains (slits, airfoils, jagged surfaces). The method of conformal mappings allows one to transform such domains into simple regions (circle, half-plane), solve the problem there, and then map the solution back. This works because conformal mappings preserve shape (angles), meaning that the Laplace equation (which determines potential) retains its form.

Conformal Mapping

A holomorphic function $f$ with $f'(z_0) \neq 0$ is conformal at the point $z_0$: it preserves the angles between curves and their orientation.

Proof. Let $\gamma_1, \gamma_2$ be curves passing through $z_0$, $\gamma_1'(0) = v_1$, $\gamma_2'(0) = v_2$. Their images: $(f\circ\gamma_i)'(0) = f'(z_0)\cdot v_i$. Multiplication by $f'(z_0)\ne 0$ rotates both vectors by the same angle $\arg f'(z_0)$, so the angle between them is preserved. The scale changes by $|f'(z_0)|$—the same in all directions.

Möbius (Fractional-Linear) Transformations

Form: $w = \frac{az+b}{cz+d}$, $ad-bc \neq 0$.

This is a bijection of the extended plane $\mathbb{C}\cup{\infty}$ onto itself (i.e., a mapping of the Riemann sphere). Each Möbius transformation is a composition of translations, rotations, scalings, and inversion $z \mapsto 1/z$.

Key properties:

• Map lines and circles to lines and circles.
• 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 determined by three pairs "point $\to$ image".

Cayley mapping: $\mathbb{H} \to \mathbb{D}$ (upper half-plane to the unit disk): $ w = \frac{z - i}{z + i}. $ Inverse: $z = i\frac{1+w}{1-w}$. The real axis maps to the unit circle, the point $i \mapsto 0$.

Riemann's Theorem

Theorem (Riemann, 1851): Any simply connected domain $D \subsetneq \mathbb{C}$ is conformally equivalent to the unit disk $\mathbb{D} = {|z| < 1}$.

This is a powerful existence theorem: a conformal isomorphism exists even when there is no explicit formula. Uniqueness: if $z_0 \in D$ is fixed with $f(z_0) = 0$ and $f'(z_0) > 0$, the mapping is unique.

Applications

Aerodynamics—the Zhukovsky airfoil: The Zhukovsky mapping $w = z + 1/z$ transforms a circle into the shape of an airplane wing (the “Zhukovsky airfoil”). The problem of flow around an airfoil in an ideal fluid reduces to the elementary problem of flow around a circle. Circulation determines lift (Kutta–Joukowski formula: $L = \rho V \Gamma$).

Electrostatics: A capacitor consisting of two eccentric cylinders $\to$ conformal mapping to concentric cylinders $\to$ capacitance can be computed explicitly.

Numerical Example

Problem: Find the image of the rectangle $R = {0 \leq \operatorname{Re} z \leq 1,, 0 \leq \operatorname{Im} z \leq 1}$ under the mapping $w = e^z$.

Step 1. Left side $z = i y$, $y \in [0,1]$: $w = e^{i y} = \cos y + i\sin y$—an arc of the unit circle from $1$ to $\cos 1 + i\sin 1$ (≈ $0.54 + 0.84i$).

Step 2. Right side $z = 1 + i y$, $y \in [0,1]$: $w = e\cdot e^{i y}$—an arc of a circle of radius $e \approx 2.718$ from $e$ to $e\cdot (\cos 1 + i\sin 1)$.

Step 3. Bottom side $z = x$, $x \in [0,1]$: $w = e^x$—a segment of the real axis from $1$ to $e$.

Step 4. Top side $z = x + i$, $x \in [0,1]$: $w = e^x e^{i} = e^x(\cos 1 + i\sin 1)$—a ray from the origin at an angle $1$ radian, from the point $\cos 1 + i\sin 1$ to $e(\cos 1 + i\sin 1)$.

Conclusion: The rectangle is mapped to a "curvilinear trapezoidal" sector, bounded by two arcs of circles and two rays. The angles at the rectangle’s vertices ($90^\circ$) are preserved—a property of conformality. Exception: $z = 0$, where $f'(0) = e^0 = 1 \neq 0$, so conformality holds.

Real-World Application

Cartography: conformal projections (Mercator, stereographic) preserve angles on the map, which is crucial for navigation—a ship's course on a Mercator map is a straight line.

Additional Aspects

Apart from the Cayley mapping, other frequently used mappings include: the power mapping $w = z^\alpha$ (opens or closes angles—e.g., a wedge of angle $\pi/n$ is mapped to the half-plane); the logarithm $w = \ln z$ (unwraps an annulus into a rectangle); Zhukovsky's mapping $w = (z + 1/z)/2$ (maps the circle to the airfoil profile—basis of classical aerodynamics). The Schwarz–Christoffel symmetrization provides an explicit formula for mapping the half-plane to any polygon via the integral of a product of powers $(z - a_k)^{\alpha_k-1}$. Numerical implementation of conformal mappings underlies packages like SC Toolbox (MATLAB) and conformalmaps (Python), which are used in modeling microfluidics, heat transfer in complex domains, and in computer graphics for seamless texture unwrapping.

Connection With Other Branches of Mathematics

Conformal mappings are closely related to problems involving the Laplace equation and, more broadly, with the theory of elliptic equations. The classical method exploits the invariance of harmonic functions under conformal bijections: solutions to boundary value problems (Dirichlet, Neumann) are transferred from a complex domain to a circle or half-plane. In an abstract theoretical context, this manifests in Liouville's theorem on the global structure of biholomorphic automorphisms.

The topological aspect is expressed in the concept of Riemann surfaces: the conformal class of metrics defines the complex structure. Here, the Poincaré–Koebe uniformization theorem applies: any simply connected Riemann surface is conformally equivalent to the sphere, the plane, or the disk. This directly connects conformal mappings to hyperbolic geometry and Fuchsian groups; fractional-linear mappings act as isometries of the Poincaré model.

In probability theory, conformal mappings are a key tool in the study of two-dimensional Brownian motion. A classical result: the image of a Brownian trajectory under a conformal map is again Brownian motion with a time change. A modern example is the stochastic conformal invariance of SLE (Schramm–Loewner Evolution) processes, where the evolution of layers of conformal mappings describes the boundaries of random fractals.

In numerical methods, conformal mappings are applied for conformal mesh generation: a complex domain is first conformally mapped to a canonical domain (circle, strip), where a regular grid is easily constructed; the mesh is then carried back. This is used in computational fluid dynamics and electromagnetics. The algebraic aspect is manifested through the action of the group $\mathrm{PSL}(2, \mathbb{C})$ on the Riemann sphere, where the combination of Möbius transformations is described by matrix multiplication; the classification theorem by the trace of the matrix (elliptic, parabolic, hyperbolic elements) connects conformal symmetries with representation theory.

Historical Background and Development of the Idea

The first systematic studies of angle-preserving mappings date back to the works of Gauss on geodesics on surfaces and his theory of cartographic projections, published in the 1820s–1830s. He already considered locally conformal parameters on surfaces and stereographic projection. Bernhard Riemann, in his 1851 dissertation, formulated and proved the theorem now bearing his name, on the conformal equivalence of simply connected domains to the disk. He was motivated by questions of potential theory and flows in the plane.