Schwarz triangle function: Difference between revisions

Revision as of 02:33, 17 August 2022

Revision as of 03:08, 17 August 2022

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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 ''α, β'', and ''γ'' canare each be expressed as the reciprocal of an integer or zero, the triangle is a [[Möbius triangle]], i.e. a non-overlapping Schwarz triangle. When the target triangle is a Möbius triangle, the inverse can be expressed as:

* Spherical: [[rational function]]

* Euclidean: [[elliptical function]]