Legendre polynomials

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

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. 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:

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

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]

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


Recursive Definition

Expanding the Taylor series in equation (1) for the first two terms gives

for the first two Legendre Polynomials. To obtain further terms without resorting to direct expansion of the Taylor series, equation (1) is differentiated with respect to t on both sides and rearranged to obtain

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