Schwarzian Derivative - Conformal Mapping of Circular Arc Polygons

Conformal Mapping of Circular Arc Polygons

The Schwarzian derivative and associated second order ordinary differential equation can be used to determine the Riemann mapping between the upper half-plane or unit circle and any bounded polygon in the complex plane, the edges of which are circular arcs or straight lines. For polygons with straight edges, this reduces to the Schwarz–Christoffel mapping, which can be derived directly without using the Schwarzian derivative. The accessory parameters that arise as constants of integration are related to the eigenvalues of the second order differential equation. Already in 1890 Felix Klein had studied the case of quadrilaterals in terms of the Lamé differential equation.

Let Δ be a circular arc polygon with angles in clockwise order. Let f(z) be a holomorphic map from the upper half-plane to Δ extending continuously to a map between the boundaries. Let the vertices correspond to points on the real axis. Then p(z)=S(f)(z) is real-valued for x real and not one of the points. By the Schwarz reflection principle p(z) extends to a rational function on the complex plane with a double pole at :

The real numbers are called accessory parameters. They are subject to 3 linear constraints:

which correspond to the vanishing of the coefficients of, and in the expansion of p(z) around z = ∞. The mapping f(z) can then be written as

where and are linearly independent holomorphic solutions of the linear second order ordinary differential equation

There are n-3 linearly independent accessory parameters, which can be difficult to determine in practise.

For a triangle, when n=3, there are no accessory parameters. The ordinary differential equation is equivalent to the hypergeometric differential equation and f(z) can be written in terms of hypergeometric functions.

For a quadrilateral the accessory parameters depend on one independent variable λ. Writing U(z)=q(z)u(z) for a suitable choice of q(z), the ordinary differential equation takes the form

Thus are eigenfunctions of a Sturm-Liouville equation on the interval . By the Sturm separation theorem, the non-vanishing of forces λ to be the lowest eigenvalue.