Schwarz triangle function: Difference between revisions

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[[File:Schwarz triangle function.svg|thumb|The upper half-plane, and the image of the upper half-plane transformed by the Schwarz triangle function with various parameters.]]

{{Complex analysis sidebar}}

In [[complex analysis]], the '''Schwarz triangle function''' or '''Schwarz s-function''' is a function that [[conformal mapping|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 case 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==

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== Inverse ==

When ''α, β'', and ''γ'' are rational, the triangle is a Schwarz triangle. When each of ''α, β'', and ''γ'' are either the reciprocal of an integer or zero, the triangle is a [[Möbius triangle]], i.e. a non-overlapping Schwarz triangle. For a Möbius triangle, the inverse is a [[single-valued]] function, which are [[automorphicmodular function]]s. for the [[triangle group]] of Möbius transformations for the given Möbius triangle.

Depending onIn the geometryspherical ofcase, thethat trianglemodular function is a [[rational function]]. For Euclidean triangles, the inverse function can be expressed using: [[elliptical function]]s.<ref name=Lee>

* Spherical: [[rational function]]s

* Euclidean: [[elliptical function]]s

* Hyperbolic: [[modular function]]s

== Ideal triangles ==