---
tag: RS8Z48E784
kind: concept
name: Riemann's differential equation
msc: ["33C05", "34M03"]
specializes: [ZBVH2YE50B]
sources:
  - url: https://en.wikipedia.org/wiki/Riemann%27s_differential_equation
same_as:
  wikidata: Q974071
  wikipedia: Riemann's differential equation
---

## Summary

Riemann's differential equation is the general second-order linear [[Fuchsian equation|ZBVH2YE50B]] with exactly three [[regular singular points|H96HXG6CA0]] on the Riemann sphere.
It is determined by the positions of the three points and the two exponents at each.
The six exponents must add up to $1$.
The [[hypergeometric differential equation|W8VE25R5A7]] is the case where the points are $0$, $1$ and $\infty$.
Every Riemann equation reduces to it by a [[Möbius transformation|0SG74WBFA2]] and a power factor.

## Explanations

### L1 human

A second-order linear equation whose solutions behave like powers near three special points, and nowhere else misbehave, is completely fixed by those three points and the six powers.
Riemann wrote down this equation explicitly.
Moving the three points to $0$, $1$, $\infty$ by a Möbius map, and multiplying the unknown by suitable powers, turns it into the hypergeometric equation.
So the theory of the hypergeometric function covers every such equation.