Legendre Polynomials (redirected from Legendre's differential equation)

Legendre polynomials [lə′zhän·drə ‚päl·i′nō·mē·əlz]

(mathematics)

A collection of orthogonal polynomials which provide solutions to the Legendre equation for nonnegative integral values of the parameter.

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Legendre Polynomials 

a system of polynomials of successively increasing degree. The polynomials were first investigated by A. Legendre and P. Laplace independently of each other between 1782 and 1785. For n = 0, 1, 2, …, the Legendre polynomials P_n(x) can be defined by the formula

The first few polynomials are

p₀ (x) = 1

p₁ (x) = x

p₂ (x) = 1/2(3x² - 1)

p₃ (x) = 1/2(5x² - 3x)

p₄ (x) = 1/8(35x⁴ - 30x² + 3)

p₅ (x) = 1/8(63x⁵ - 70x³ + 15x)

All the zeros of P _n (x) are real, lie in the interval [–1, + 1], and alternate with the zeros of P_{n + 1} (x). The Legendre polynomials are a complete set of orthogonal polynomials on the interval [–1, + 1]. Thus, it is possible to expand an arbitrary function /(jc) integrable over the interval [– 1, +1] in a series of Legendre polynomials:

where

The type of convergence of this series is roughly the same as that of a Fourier series. The Legendre polynomials are given explicitly by the formula

The generating function is

that is, the Legendre polynomials are the coefficients in the expansion of this function in powers of t. They are recursively defined by

nP_n (x) + (n - 1)P_n-2 (x) - (2n - 1)xP_n-1 (x) = 0

P_n (x)satisfies the differential equation

which arises when separating the variables in Laplace’s equation in spherical coordinates.

REFERENCES

Janke, E., F. Emde, and F. Lösch. Spetsial’nye funktsii; grafiki, tablitsy, 2nd ed. Moscow, 1968. (Translated from German.)
Lebedev, N. N. Spetsial’nye funktsii i ikh prilozheniia, 2nd ed. MoscowLeningrad, 1963.

V. I. BITIUTSKOV