---
tag: W6VE30V10Y
kind: example
name: Legendre's equation
msc: [34B30, 33C45, 34M03]
instance_of: [ZBVH2YE50B]
sources:
  - url: https://en.wikipedia.org/wiki/Legendre_polynomials
---

## Example

Let $\lambda \in \C$. *Legendre's equation* is
$$(1 - z^2)\, y'' - 2 z\, y' + \lambda\, y = 0,$$
usually written with $\lambda = n (n + 1)$.
In normalised form, $p(z) = \frac{-2z}{1 - z^2}$ and $q(z) = \frac{\lambda}{1 - z^2}$, which are rational.
They have simple poles at $z = \pm 1$, so $\pm 1$ are regular singular points, and every other finite point is ordinary.
As $z \to \infty$, $z p(z) \to 2$ and $z^2 q(z) \to -\lambda$, so $\infty$ is ordinary or regular singular.
Hence Legendre's equation is a [[Fuchsian equation|ZBVH2YE50B]], with singular points among $-1$, $1$ and $\infty$.

## Remarks

At the [[ordinary point|QK2J038VB4]] $0$, a [[power series solution|08RT9GDDPQ]] $\sum a_k z^k$ satisfies $(k + 2)(k + 1)\, a_{k+2} = \left( k (k + 1) - \lambda \right) a_k$.
The series converge for $\abs{z} < 1$, the distance from $0$ to the nearest singular point.
If $\lambda = n (n + 1)$ with $n \ge 0$ an integer, the recurrence stops, and one solution is the Legendre polynomial $P_n$ of degree $n$.
For $n = 2$, $P_2(z) = \frac{3 z^2 - 1}{2}$: then $y'' = 3$ and $y' = 3z$, and $(1 - z^2) \cdot 3 - 2z \cdot 3z + 6 \cdot \frac{3 z^2 - 1}{2} = 0$.
The equation arises from the Laplace equation in spherical coordinates.

## Explanations

### L1 human

Legendre's equation appears when one studies gravity or electric potential around a sphere.
Its only bad points are the poles $z = \pm 1$ of the sphere and the point at infinity, and all of them are tame.
For special values of the parameter it has polynomial solutions, the Legendre polynomials.