Schwarz triangle function

In mathematics, the Schwarz triangle function was introduced by H. A. Schwarz as the inverse function of the conformal mapping uniformizing a Schwarz triangle, i.e. a geodesic triangle in the upper half plane with angles which are either 0 or of the form π over a positive integer greater then one. Applying successive hyperbolic reflections in its sides, such a triangle generates a tessellation of the upper half plane (or the unit disk after composition with the Cayley transform). The conformal mapping of the upper half plane onto the interior of the geodesic triangle generalizes the Schwarz-Christoffel transformation. Through the theory of the Schwarzian derivative, it can be expressed as the quotient of two solutions of a hypergeometric differential equation. By the Schwarz reflection principle and monodromy properties of the ordinary differential equation, the inverse function is an automorphic function for the orientation-preserving subgroup of the discrete group generated by hyperbolic reflections in the sides of the triangle. This is a special case of a general method of Henri Poincaré that associates automorphic forms with ordinary differential equations with regular singular points. In the special case of ideal triangles, where all the angles are zero, the tessellation corresponds to the Farey tessellation and the triangle functions yield the modular lambda function and elliptic modular function.

References

• Caratheodory, C. (1954), Theory of functions of a complex variable. Vol. 2., translated by F. Steinhardt., Chelsea Publishing Company
• Ford, Lester R. (1951), Automorphic Functions, American Mathematical Society, ISBN 0821837419 {{citation}}: ISBN / Date incompatibility (help), reprint of 1929 edition
• Helgason, Sigurdur (2000), Groups and geometric analysis. Integral geometry, invariant differential operators, and spherical functions, Mathematical Surveys and Monographs, vol. 83, American Mathematical Society, ISBN 0-8218-2673-5
• Hille, Einar (1976), Ordinary differential equations in the complex domain, Wiley-Interscience
• Ince, E. L. (1944), Ordinary Differential Equations, Dover Publications
• Magnus, Wilhelm (1974), Noneuclidean tesselations and their groups, Pure and Applied Mathematics, vol. 61, Academic Press
• Nehari, Zeev (1975), Conformal mapping, Dover Publications
• Thurston, William P. (1997), Silvio Levy (ed.), Three-dimensional geometry and topology. Vol. 1., Princeton Mathematical Series, vol. 35, Princeton University Press, ISBN 0-691-08304-5
• Wolf, Joseph A. (2011), Spaces of constant curvature (Sixth ed.), AMS Chelsea Publishing, ISBN 978-0-8218-5282-8