---
tag: ZBVH2YE50B
kind: concept
name: Fuchsian equation
msc: [34M03, 34M35]
sources:
  - url: https://en.wikipedia.org/wiki/Fuchsian_theory
---

## Summary

A Fuchsian equation is a linear differential equation with rational coefficients whose singular points are all [[regular singular points|H96HXG6CA0]].
This includes the point $\infty$ of the Riemann sphere.
Its solutions are multivalued functions of controlled growth, described near each singular point by [[Fuchs' theorem|6EZ400WM94]].
The equation is largely determined by its singular points, its exponents and its [[monodromy|C8HD62AB6Q]].
The hypergeometric equation, the Legendre equation and the Riemann equation are Fuchsian.
See the [[definition|YT2B9KQ0BG]].

## Explanations

### L1 human

A Fuchsian equation has finitely many bad points, and at each of them the solutions only behave like powers and logarithms.
Nothing wild happens anywhere, not even at infinity.
The Gauss hypergeometric equation is the main example, with bad points $0$, $1$ and $\infty$.
The Bessel equation is not Fuchsian, because its solutions oscillate like $\sin z$ near $\infty$.