Schwarz–Christoffel mapping
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 andHermann Amandus Schwarz.
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[edit]Definition
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
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 
, then this mapping is given by
where K is a constant, and a < b < c < ... are the values, along the real axis of the ζ 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 ζ plane maps to one of the vertices of the z plane polygon (conventionally the vertex with angle α). 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.
[edit]Example
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 = π⁄2 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
Evaluation of this integral yields
- z> = f(ζ) = C + K ar cosh 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 = ar cosh ζ
This transformation is sketched below.
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Other simple mappings
[edit]Triangle
A mapping to a plane triangle with angles 
and π(1 − a − b) is given by
[edit]Square
The upper half-plane is mapped to the square by
where F is the incomplete elliptic integral of the first kind.
[edit]General triangle
The upper half-plane is mapped to a triangle with circular arcs for edges by the Schwarz triangle map.
[edit]See also
- The Schwarzian derivative appears in the theory of Schwarz–Christoffel mappings.
[edit]References
- Driscoll, Tobin A.; Trefethen, Lloyd N. (2002), Schwarz-Christoffel mapping, Cambridge Monographs on Applied and Computational Mathematics, 8, Cambridge University Press, MR1908657, ISBN 978-0-521-80726-5
- Nehari, Zeev (1982) [1952], Conformal mapping, New York: Dover Publications, MR0045823, ISBN 978-0-486-61137-2
[edit]Further reading
- Case, James (2008), "Breakthrough in Conformal Mapping", SIAM News 41 (1).
