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.
It is often convenient to consider the case in which the point at infinity of the plane maps to one of the vertices of the plane polygon (conventionally the vertex with angle ). If this is done, the first factor in the formula is effectively a constant and may be regarded as being absorbed into the constant .
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 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.
Other simple mappings
A mapping to a plane triangle with angles and is given by
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, 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
- Case, James (2008), "Breakthrough in Conformal Mapping" (PDF), SIAM News 41 (1).