## Dr. Ahmed G. Abo-Khalil

Electrical Engineering Department

# Legendre polynomials

In mathematics, Legendre functions are solutions to Legendre's differential equation:

${d over dx} left[ (1-x^2) {d over dx} P_n(x) ight] + n(n+1)P_n(x) = 0.$

(1)

They are named after Adrien-Marie Legendre. This ordinary differential equation is frequently encountered in physics and other technical fields. In particular, it occurs when solving Laplace's equation (and related partial differential equations) in spherical coordinates.

The Legendre differential equation may be solved using the standard power series method. The equation has regular singular points at x = ±1 so, in general, a series solution about the origin will only converge for |x| < 1. When n is an integer, the solution Pn(x) that is regular at x = 1 is also regular at x = −1, and the series for this solution terminates (i.e. it is a polynomial).

These solutions for n = 0, 1, 2, ... (with the normalization Pn(1) = 1) form a polynomial sequence of orthogonal polynomials called the Legendre polynomials. Each Legendre polynomial Pn(x) is an nth-degree polynomial. It may be expressed using Rodrigues' formula:

$P_n(x) = {1 over 2^n n!} {d^n over dx^n } left[ (x^2 -1)^n ight].$

That these polynomials satisfy the Legendre differential equation (1) follows by differentiating n + 1 times both sides of the identity

$(x^2-1)frac{d}{dx}(x^2-1)^n = 2nx(x^2-1)^n$

and employing the general Leibniz rule for repeated differentiation.[1] The Pn can also be defined as the coefficients in a Taylor series expansion:[2]

$frac{1}{sqrt{1-2xt+t^2}} = sum_{n=0}^infty P_n(x) t^n.$

(2)

In physics, this ordinary generating function is the basis for multipole expansions.

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