Schwarz triangle function
Mathematical analysis → Complex analysis |
Complex analysis |
---|
Complex numbers |
Complex functions |
Basic theory |
Geometric function theory |
People |
In complex analysis, the Schwarz triangle function or Schwarz s-function is a function that conformally maps the upper half plane to a triangle in the upper half plane having lines or circular arcs for edges. The target triangle is not necessarily a Schwarz triangle, although that is the most mathematically interesting case. When that triangle is a non-overlapping Schwarz triangle, i.e. a Möbius triangle, the inverse of the Schwarz triangle function is a single-valued automorphic function for that triangle's triangle group. More specifically, it is a modular function.
Formula[edit]
Let
where
- a = (1−
α −β −γ )/2, - b = (1−
α +β −γ )/2, - c = 1−
α , - a′ = a − c + 1 = (1+
α −β −γ )/2, - b′ = b − c + 1 = (1+
α +β −γ )/2, and - c′ = 2 − c = 1 +
α .
This function maps the upper half-plane to a spherical triangle if
Derivation[edit]
Through the theory of complex ordinary differential equations with regular singular points and the Schwarzian derivative, the triangle function can be expressed as the quotient of two solutions of a hypergeometric differential equation with real coefficients and singular points at 0, 1 and ∞. By the Schwarz reflection principle, the reflection group induces an action on the two dimensional space of solutions. On the orientation-preserving normal subgroup, this two-dimensional representation corresponds to the monodromy of the ordinary differential equation and induces a group of Möbius transformations on quotients of hypergeometric functions.[2]
Singular points[edit]
This mapping has regular singular points at z = 0, 1, and ∞, corresponding to the vertices of the triangle with angles
where is the gamma function.
Near each singular point, the function may be approximated as
where is big O notation.
Inverse[edit]
When
In the spherical case, that modular function is a rational function. For Euclidean triangles, the inverse can be expressed using elliptical functions.[4]
Ideal triangles[edit]
When
Instead, a mapping to an ideal triangle with vertices at 0, 1, and ∞ is given by in terms of the complete elliptic integral of the first kind:
- .
This expression is the inverse of the modular lambda function.[5]
Extensions[edit]
The Schwarz–Christoffel transformation gives the mapping from the upper half-plane to any Euclidean polygon.
The methodology used to derive the Schwarz triangle function earlier can be applied more generally to arc-edged polygons. However, for an n-sided polygon, the solution has n-3 additional parameters, which are difficult to determine in practice.[6] See Schwarzian derivative § Conformal mapping of circular arc polygons for more details.
Applications[edit]
L. P. Lee used Schwarz triangle functions to derive conformal map projections onto polyhedral surfaces.[4]
References[edit]
- ^ Nehari 1975, p. 309.
- ^ Nehari 1975, pp. 198–208.
- ^ Nehari 1975, pp. 315−316.
- ^ a b Lee, Laurence (1976). Conformal Projections based on Elliptic Functions. Cartographica Monographs. Vol. 16. University of Toronto Press. ISBN 9780919870161. Chapters also published in The Canadian Cartographer. 13 (1). 1976.
- ^ Nehari 1975, pp. 316–318.
- ^ Nehari 1975, p. 202.
Sources[edit]
- Ahlfors, Lars V. (1979). Complex analysis: an introduction to the theory of analytic functions of one complex variable (3 ed.). New York: McGraw-Hill. ISBN 0-07-000657-1. OCLC 4036464.
- Carathéodory, Constantin (1954). Theory of functions of a complex variable. Vol. 2. Translated by F. Steinhardt. Chelsea. OCLC 926250115.
- Hille, Einar (1997). Ordinary differential equations in the complex domain. Mineola, N.Y.: Dover Publications. ISBN 0-486-69620-0. OCLC 36225146.
- Nehari, Zeev (1975). Conformal mapping. New York: Dover Publications. ISBN 0-486-61137-X. OCLC 1504503.
- Sansone, Giovanni; Gerretsen, Johan (1969). Lectures on the theory of functions of a complex variable. II: Geometric theory. Wolters-Noordhoff. OCLC 245996162.