---
tag: XMD3Q49Z2C
kind: concept
name: Schwarz triangle function
msc: ["33C05", "30C20"]
sources:
  - url: https://en.wikipedia.org/wiki/Schwarz_triangle_function
same_as:
  wikidata: Q30324144
  wikipedia: Schwarz triangle function
---

## Summary

The Schwarz triangle function is the ratio of two independent solutions of the [[hypergeometric differential equation|W8VE25R5A7]].
For suitable real parameters it is a [[conformal map|0YYT36R4YD]] 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.

## Explanations

### L1 human

A ratio of two solutions of a second-order linear equation forgets the scaling of the solutions and keeps only their shape.
For the hypergeometric equation, this ratio maps the upper half-plane onto a curved triangle, whose sides are arcs of circles.
The three vertices are the images of $0$, $1$ and $\infty$, and the angles there are fixed by the parameters $a$, $b$, $c$.
When the angles are of the form $\pi/n$, reflecting the triangle across its sides again and again tiles a disc, the plane or the sphere.
The case of the sphere is how Schwarz found the algebraic hypergeometric functions.