Schwarz–Christoffel mapping

From Wikipedia, the free encyclopedia
Jump to: navigation, search

In complex analysis, a Schwarz–Christoffel mapping is a conformal transformation of the upper half-plane onto the interior of a simple polygon. Schwarz–Christoffel mappings are used in potential theory and some of its applications, including minimal surfaces and fluid dynamics. They are named after Elwin Bruno Christoffel and Hermann Amandus Schwarz.

Definition[edit]

Consider a polygon in the complex plane. The Riemann mapping theorem implies that there is a bijective biholomorphic mapping f from the upper half-plane

 \{ \zeta \in \mathbb{C}: \operatorname{Im}\,\zeta > 0 \}

to the interior of the polygon. The function f maps the real axis to the edges of the polygon. If the polygon has interior angles \alpha,\beta,\gamma, \ldots, then this mapping is given by


f(\zeta) = \int^\zeta \frac{K}{(w-a)^{1-(\alpha/\pi)}(w-b)^{1-(\beta/\pi)}(w-c)^{1-(\gamma/\pi)} \cdots} \,\mbox{d}w

where K is a constant, and a < b < c < ... are the values, along the real axis of the \zeta plane, of points corresponding to the vertices of the polygon in the z plane. A transformation of this form is called a Schwarz–Christoffel mapping.

It is often convenient to consider the case in which the point at infinity of the \zeta plane maps to one of the vertices of the z plane polygon (conventionally the vertex with angle \alpha). If this is done, the first factor in the formula is effectively a constant and may be regarded as being absorbed into the constant K.

Example[edit]

Consider a semi-infinite strip in the z plane. This may be regarded as a limiting form of a triangle with vertices P = 0, Q = π i, and R (with R real), as R tends to infinity. Now α = 0 and β = γ = π2 in the limit. Suppose we are looking for the mapping f with f(−1) = Q, f(1) = P, and f(∞) = R. Then f is given by

 f(\zeta) = \int^\zeta 
  \frac{K}{(w-1)^{1/2}(w+1)^{1/2}} \,\mbox{d}w. \,

Evaluation of this integral yields

z = f(ζ) = C + K arccosh ζ

where C is a (complex) constant of integration. Requiring that f(−1) = Q and f(1) = P gives C = 0 and K = 1. Hence the Schwarz–Christoffel mapping is given by

z = arccosh ζ

This transformation is sketched below.

Schwarz–Christoffel mapping of the upper half-plane to the semi-infinite strip

Other simple mappings[edit]

Triangle[edit]

A mapping to a plane triangle with angles \pi a,\, \pi b and \pi(1-a-b) is given by

z=f(\zeta)=\int^\zeta \frac{dw}{(w-1)^{1-a} (w+1)^{1-b}}.

Square[edit]

The upper half-plane is mapped to the square by

z=f(\zeta) = \int^\zeta \frac {\mbox{d}w}{\sqrt{w(w^2-1)}}
=\sqrt{2} \, F\left(\sqrt{\zeta+1};\sqrt{2}/2\right),

where F is the incomplete elliptic integral of the first kind.

General triangle[edit]

The upper half-plane is mapped to a triangle with circular arcs for edges by the Schwarz triangle map.

See also[edit]

References[edit]

Further reading[edit]

External links[edit]