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.
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 is a constant, and are the values, along the real axis of the plane, of points corresponding to the vertices of the polygon in the plane. A transformation of this form is called a Schwarz–Christoffel mapping.
The integral can be simplified by mapping the point at infinity of the plane to one of the vertices of the plane polygon. By doing this, the first factor in the formula becomes constant and so can be absorbed into the constant . Conventionally, the point at infinity would be mapped to the vertex with angle .
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
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
This transformation is sketched below.
Other simple mappings
A mapping to a plane triangle with interior angles and is given by
which can be expressed in terms of hypergeometric functions.
The upper half-plane is mapped to the square by
where F is the incomplete elliptic integral of the first kind.
The upper half-plane is mapped to a triangle with circular arcs for edges by the Schwarz triangle map.
- The Schwarzian derivative appears in the theory of Schwarz–Christoffel mappings.
- Driscoll, Tobin A.; Trefethen, Lloyd N. (2002), Schwarz–Christoffel mapping, Cambridge Monographs on Applied and Computational Mathematics, 8, Cambridge University Press, doi:10.1017/CBO9780511546808, ISBN 978-0-521-80726-5, MR 1908657
- Nehari, Zeev (1982) , Conformal mapping, New York: Dover Publications, ISBN 978-0-486-61137-2, MR 0045823
- The Conformal Hyperbolic Square and Its Ilk Chamberlain Fong, Bridges Finland Conference Proceedings, 2016
An analogue of SC mapping that works also for multiply-connected is presented in: Case, James (2008), "Breakthrough in Conformal Mapping" (PDF), SIAM News, 41 (1).