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]]):. Each of ''α'', ''β'', and ''γ'' may take values between 0 and 1 inclusive. followingFollowing {{harv|Nehari|1975|page=309}}, thisthese angles are in clockwise order., Ifwith anythe ofvertex having angle ''α, βπα'', at the origin and the vertex having angle ''γπγ'' arelying greateron thanthe zero,real thenline. theThe 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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:''b''′ = ''b'' − ''c'' + 1 = (1+α+β−γ)/2, and

:''c''′ = 2 − ''c'' = 1 + α.

This formula can be derived using the [[Schwarzian derivative]].

This function maps the upper half-plane to a [[spherical triangle]] if α + β + γ > 1, or a [[hyperbolic triangle]] if α + β + γ < 1. When α + β + γ = 1, then the triangle is a Euclidean triangle with straight edges: ''a''&thinsp;=&thinsp;0, <math>_2 F_1 \left(a, b; c; z\right) = 1</math>, and the formula reduces to that given by the [[Schwarz–Christoffel transformation]]. In the special case of [[ideal triangle]]s, where all the angles are zero, the triangle function yields the [[modular lambda function]].

When ''α'' = 0 the triangle is degenerate, lying entirely on the real line. If either of ''β'' or ''γ'' are non-zero, the angles can be permuted so that the positive value is ''α''. For an [[ideal triangle]] having all angles zero, a mapping to a triangle with vertices at ''i'', 1, and ''-i'' is given by a transformed function with ''α''=''β''=0, ''γ''= 1/2, and <math>z = \frac{1}{1-w^2}</math>.{{sfn|Nehari|1975|pp=316-317}}

===Derivation===

This formula can be derived using the [[Schwarzian derivative]].

== Singular points ==