Schwarz triangle function: Difference between revisions

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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]] with real coefficients and singular points at 0, 1 and ∞. By the [[Schwarz reflection principle]], andthe monodromydiscrete propertiesgroup ofgenerated theby ordinaryhyperbolic differentialreflections equation,in the inversesides functionof isthe triangle induces an [[automorphicaction function]]on forthe two dimensional space of solutions. On the orientation-preserving normal subgroup, this two dimensional representation corresponds to the [[monodromy]] of the discreteordinary groupdifferential generatedequation byand hyperbolicinduces reflectionsa ingroup theof sides[[Möbius transformation]]s on quotients of solutions. Since the triangle function is the inverse function of such a quotient, it is therefore an [[automorphic function]] for this discrete group of Möbius transformations. This is a special case of a general method of [[Henri Poincaré]] that associates automorphic forms with [[ordinary differential equation]]s with [[regular singular point]]s. In the special case of [[ideal triangle]]s, where all the angles are zero, the tessellation corresponds to the [[Farey series|Farey tessellation]] and the triangle functionsfunction yieldyields the [[modular lambda function]]. and [[j-invariant|elliptic modular function]].

==References==

*{{citation|last=Caratheodory|first=C.|authorlink=Constantin Caratheodory|title=Theory of functions of a complex variable. Vol. 2.|translator=F. Steinhardt. |publisher=Chelsea Publishing Company|year= 1954}}