Ordinary differential equations Complex domain and special equations Example

Legendre's equation

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, with singular points among $-1$ , $1$ and $\infty$ .

Remarks

At the ordinary point $0$ , a power series solution $\sum a_k z^k$ 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.

Tag	`W6VE30V10Y`
URI	`m2://W6VE30V10Y`
Kind	example
Writing	W0 stub
MSC	34B30, 33C45, 34M03
Instance of	Fuchsian equation
Changed	2 November 2026

Legendre's equation

Example

Remarks

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