In mathematics, Legendre functions are solutions to Legendre's differential equation:
They are named after AdrienMarie 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 P_{n}(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 P_{n}(1) = 1) form a polynomial sequence of orthogonal polynomials called the Legendre polynomials. Each Legendre polynomial P_{n}(x) is an nthdegree 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 P_{n} can also be defined as the coefficients in a Taylor series expansion:^{[2]}
In physics, this generating function is the basis for multipole expansions.
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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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