Schwarz-Christoffel Mapping

The Schwarz-Christoffel formula provides an explicit conformal map from the upper half-plane (or the unit disk) to the interior of a polygon. This is one of the most useful explicit constructions in applied complex analysis.

The Schwarz-Christoffel Formula

Theorem7.2Schwarz-Christoffel formula

Let $P$ be a polygon with vertices $w_1, \ldots, w_n$ (in order) and interior angles $\alpha_1\pi, \ldots, \alpha_n\pi$ (where $\sum \alpha_k = n - 2$ ). The conformal map $f: \mathbb{H} \to \text{Int}(P)$ from the upper half-plane to the interior of $P$ has the form

$f(z) = A \int_{z_0}^z \prod_{k=1}^{n} (\zeta - x_k)^{\alpha_k - 1}\,d\zeta + B$

where $x_1 < x_2 < \cdots < x_n$ are the prevertices on $\mathbb{R}$ (with $f(x_k) = w_k$ ), and $A, B$ are constants.

RemarkDegrees of freedom

Three of the prevertices $x_k$ can be chosen freely (by the three-parameter Mobius group of $\mathbb{H}$ ). The remaining $n - 3$ prevertices and the constants $A, B$ must be determined from the polygon's geometry. This is the parameter problem, which is generally solved numerically.

Examples

ExampleMap to a rectangle

For a rectangle with vertices $\pm K$ and $\pm K + iK'$ , the interior angles are all $\pi/2$ , so $\alpha_k = 1/2$ for $k = 1,2,3,4$ . The Schwarz-Christoffel map becomes

$f(z) = A\int_0^z \frac{d\zeta}{\sqrt{(\zeta^2-1)(\zeta^2-k^2)}} = A\,\text{sn}^{-1}(z, k)$

where $\text{sn}$ is the Jacobi elliptic sine function and $k$ is the elliptic modulus. This connects conformal mapping to elliptic integrals.

ExampleMap to a half-infinite strip

The map $f(z) = \frac{1}{\pi}\log\frac{1+z}{1-z}$ sends the upper half-plane to the infinite strip $\{w : 0 < \text{Im}(w) < 1\}$ . This can be derived as a limiting case of the Schwarz-Christoffel formula where two vertices go to infinity.

The Derivative Form

Definition7.3Schwarz-Christoffel derivative

The derivative of the Schwarz-Christoffel map takes the simpler form:

$f'(z) = A \prod_{k=1}^n (z - x_k)^{\alpha_k - 1}.$

This shows that $f'$ has algebraic singularities at the prevertices. The exponent $\alpha_k - 1$ corresponds to the turning angle at vertex $w_k$ : the exterior angle is $(1 - \alpha_k)\pi$ .

ExampleEquilateral triangle

For an equilateral triangle, all angles are $\pi/3$ , so $\alpha_k = 1/3$ . Choosing prevertices at $0, 1, \infty$ :

$f'(z) = A z^{-2/3}(z-1)^{-2/3}.$

The map involves the hypergeometric function: $f(z) = A\,z^{1/3}\,{}_2F_1(1/3, 2/3; 4/3; z)$ .

Applications

RemarkPhysical applications

Schwarz-Christoffel mappings are widely used in:

Fluid dynamics: Mapping potential flow around polygonal obstacles. The complex potential in a polygon is computed by mapping to the half-plane where the solution is known.
Electrostatics: Finding the electric field in capacitors with polygonal cross-sections.
Heat conduction: Solving Laplace's equation in polygonal domains via conformal transformation.
Aerodynamics: The Joukowski airfoil is a special case; general airfoil shapes use Schwarz-Christoffel ideas.

RemarkNumerical computation

The Schwarz-Christoffel Toolbox (by Driscoll) provides efficient numerical computation of these maps. The main challenge is solving the nonlinear system of equations for the prevertices, which requires Newton's method initialized with good starting values.