Schwarz triangle function: Difference between revisions

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==Formula==

Let ''πα'', ''πβ'', and ''πγ'' be the interior angles at the vertices of the triangle (in [[radians]]): following Nehari, this are in clockwise order. If any of ''α, β'', and ''γ'' are greater than zero, then the Schwarz triangle function can be given in terms of [[hypergeometric functions]] as:

:<math>s(z) = z^{\alpha} \frac{_2 F_1 \left(a', b'; c'; z\right)}{_2 F_1 \left(a, b; c; z\right)}</math>

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:<math>s_0(z)=z^\alpha (1+O(z))</math>,

:<math>s_1(z)=(1-z)^\betagamma (1+O(1-z))</math>, and

:<math>s_\infty(z)=z^\gammabeta (1+O(\tfrac{1}{z}))</math>.

== Inverse ==

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 each either 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]]

* Hyperbolic: [[modular function]]

== See alsoExtensions ==

The [[Schwarz–Christoffel transformation]] gives the mapping from the upper half-plane to any Euclidean polygon.

* [[Conformal map projection]]s.<ref name=Lee>{{cite book |last=Lee |first=Laurence |title=Conformal Projections based on Elliptic Functions |year=1976 |publisher=University of Toronto Press |series=Cartographica Monographs |volume=16

The methodology used to derive the Schwarz triangle function earlier can be applied more generally to arc-edged polygons. However, for an ''n''-sided polygon, the solution has ''n-3'' additional parameters, which are difficult to determine in practice. See {{slink|Schwarzian derivative#Conformal mapping of circular arc polygons}} for more details.

== Applications ==

*L. P. Lee used Schwarz triangle functions to derive [[Conformalpolyhedral map projection]]s that were also [[conformal map projection|conformal]].<ref name=Lee>{{cite book |last=Lee |first=Laurence |title=Conformal Projections based on Elliptic Functions |year=1976 |publisher=University of Toronto Press |series=Cartographica Monographs |volume=16

|isbn=9780919870161 |url=https://archive.org/details/conformalproject0000leel |url-access=limited }} Chapters also published in [https://www.utpjournals.press/toc/cart/13/1 ''The Canadian Cartographer''. '''13''' (1). 1976.]</ref>