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.

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 ∞.

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.

[2] 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.

Instead, a mapping to an ideal triangle with vertices at 0, 1, and ∞ is given by in terms of the complete elliptic integral of the first kind: This expression is the inverse of the modular lambda function.

The methodology used to derive the Schwarz triangle function earlier can be applied more generally to arc-edged polygons.

[6] See Schwarzian derivative § Conformal mapping of circular arc polygons for more details.

L. P. Lee used Schwarz triangle functions to derive conformal map projections onto polyhedral surfaces.