# Legendre polynomial

The *Legendre polynomials ^{}* are a set of polynomials ${\{{P}_{n}\}}_{n=0}^{\mathrm{\infty}}$ each of order $n$ that satisfy Legendre’s ODE:

$$\frac{d}{dx}[(1-{x}^{2}){P}_{n}^{\prime}(x)]+n(n+1){P}_{n}(x)=0.$$ |

Alternatively ${P}_{n}$ is an eigenfunction^{} of the self-adjoint^{} differential operator $\frac{d}{dx}(1-{x}^{2})\frac{d}{dx}$ with eigenvalue^{} $-n(n+1)$.

The Legendre polynomials are also known as Legendre functions of the first kind.

By Sturm-Liouville theory, this means they’re orthogonal^{} over some interval with
some weight function. In fact it can be shown that they’re orthogonal on $[-1,1]$ with weight function $W(x)=1$. As with any set of orthogonal polynomials, this can be used to generate them (up to normalization) by Gram-Schmidt orthogonalization^{} of the monomials $\{{x}^{i}\}$. The normalization used
is $\u27e8{P}_{n}\parallel {P}_{n}\u27e9=2/(2n+1)$, which makes ${P}_{n}(\pm 1)={(\pm 1)}^{n}$

Rodrigues’s Formula (which can be generalized to some other polynomial sets) is a sometimes convenient form of ${P}_{n}$ in terms of derivatives:

$${P}_{n}(x)=\frac{1}{{2}^{n}n!}{\left(\frac{d}{dx}\right)}^{n}{({x}^{2}-1)}^{n}$$ |

The first few explicitly are:

${P}_{0}(x)$ | $=$ | $1$ | ||

${P}_{1}(x)$ | $=$ | $x$ | ||

${P}_{2}(x)$ | $=$ | $\frac{1}{2}}(3{x}^{2}-1)$ | ||

${P}_{3}(x)$ | $=$ | $\frac{1}{2}}(5{x}^{3}-3x)$ | ||

${P}_{4}(x)$ | $=$ | $\frac{1}{8}}(35{x}^{4}-30{x}^{2}+3)$ | ||

$\mathrm{\dots}$ |

As all orthogonal polynomials do, these satisfy a three-term recurrence relation:

$$(n+1){P}_{n+1}(x)=(2n+1)x{P}_{n}(x)-(n){P}_{n-1}(x)$$ |

The Legendre functions of the second kind also satisfy the Legendre ODE but are not regular at the origin.

Related are the associated Legendre functions, and spherical harmonics^{}.

Title | Legendre polynomial |
---|---|

Canonical name | LegendrePolynomial |

Date of creation | 2013-03-22 15:12:10 |

Last modified on | 2013-03-22 15:12:10 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 14 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 33C45 |

Related topic | OrthogonalPolynomials |

Defines | Rodrigues’s Formula |

Defines | Legendre’s Differential Equation^{} |