Schwarz triangle function

In complex analysis, the Schwarz triangle function or Schwarz s-function is a function that 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.

Through the theory of complex ordinary differential equations with regular singular points and the Schwarzian derivative, the triangle function can be expressed as the quotient of two solutions of a hypergeometric differential equation with real coefficients and singular points at 0, 1 and ∞. By the Schwarz reflection principle, the reflection group induces an action on the two dimensional space of solutions. On the orientation-preserving normal subgroup, this two-dimensional representation corresponds to the monodromy of the ordinary differential equation and induces a group of Möbius transformations on quotients of hypergeometric functions.

Formula

Let πα, πβ, and πγ be the interior angles at the vertices of the triangle (in radians). 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:

s(z)=z^{\alpha }{\frac {_{2}F_{1}\left(a',b';c';z\right)}{_{2}F_{1}\left(a,b;c;z\right)}}

where

a = (1−α−β−γ)/2,

b = (1−α+β−γ)/2,

c = 1−α,

a′ = a − c + 1 = (1+α−β−γ)/2,

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 = 0, $_{2}F_{1}\left(a,b;c;z\right)=1$ , and the formula reduces to that given by the Schwarz–Christoffel transformation. In the special case of ideal triangles, where all the angles are zero, the triangle function yields the modular lambda function.

Value at singular points

This mapping has regular singular points at z = 0, 1, and ∞, corresponding to the vertices of the triangle with angles πα, πγ, and πβ respectively. At these singular points,^[1]

s(0)=0

,

s(1)={\frac {\Gamma (1-a')\Gamma (1-b')\Gamma (c')}{\Gamma (1-a)\Gamma (1-b)\Gamma (c)}}

, and

s(\infty )=\exp \left(i\pi \alpha \right){\frac {\Gamma (1-a')\Gamma (b)\Gamma (c')}{\Gamma (1-a)\Gamma (b')\Gamma (c)}}

,

where Γ(x) is the Gamma function.

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 α, β, and γ are each 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

References

^ Nehari 1975, pp. 315−316.
^ Lee, Laurence (1976). Conformal Projections based on Elliptic Functions. Cartographica Monographs. Vol. 16. University of Toronto Press. ISBN 9780919870161. Chapters also published in The Canadian Cartographer. 13 (1). 1976.

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