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{{No footnotes|date=February 2017}}
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 [[hyperbolic triangle|geodesic triangle]] in the [[upper half plane]] with angles which are either 0 or of the form {{pi}} over a positive integer greater than 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]], the discrete group generated by hyperbolic reflections in the sides of the triangle 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 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 function yields the [[modular lambda function]].
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