Schwarz triangle function: Difference between revisions

Revision as of 03:05, 3 September 2022

Revision as of 03:16, 3 September 2022

Line 53:

The inverse function is an [[automorphic function]] for this discrete group of Möbius transformations. This is a special case of a general scheme of [[Henri Poincaré]] that associates automorphic forms with ordinary differential equations with regular singular points.

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. WhenFor thea targetMöbius triangle, the inverse is a Möbius[[single-valued]] trianglefunction, the inversewhich can be expressed asusing:

* Spherical: [[rational function]]s

* Euclidean: [[elliptical function]]s

* Hyperbolic: [[modular function]]s

** Ideal triangle: [[modular lambda function]]

As mentioned earlier, the Schwarz triangle function is not directly useful for ideal triangles, but the [[modular lambda function]] can be used to express maps from ideal triangles to the upper half-plane.

== Extensions ==