Schwarz triangle function

Summary

The Schwarz triangle function is the ratio of two independent solutions of the hypergeometric differential equation. For suitable real parameters it is a conformal map from the upper half-plane onto a triangle bounded by circular arcs or line segments. The angles are $\pi\lambda$, $\pi\mu$, $\pi\nu$, where $\lambda$, $\mu$, $\nu$ are the exponent differences. Its inverse functions, when the angles are $\pi/p$, $\pi/q$, $\pi/r$, are automorphic functions for triangle groups. When all three angles are zero, the inverse function is the modular lambda function.