A-to-Z Guide to Thermodynamics,
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The Legendre polynomial series is the decomposition of a function f(x) square integrable on the interval (−1, 1) into a series according to the Legendre polynomial system Pn(x) n = 0, 1,


the coefficients of series cn are calculated by:


The Legendre polynomial system is orthogonal to weight 1 and is complete on the interval (−1, 1).

The orthogonality relationships for Legendre polynomials have the form


Therefore, the Legendre polynomial series is a type of Fourier Series written in the system of orthogonal polynomials. The partial sums of a Legendre series bring the functions f(x) closer in the sense of a root-mean-square deviation and the condition limn→ ∞ cn = 0 is satisfied.

The Fourier series, written in the form of Legendre polynomials within the interval (−1, 1), is similar to the trigonometric Fourier series: convergence of both series takes place, which means that a Fourier-Legendre function f(x) converges at the points x (−1, 1) when a trigonometric series of a Fourier function F(θ) = f(cos θ) converges at the point θ = arcsin x.

Near the ends of the orthogonality interval, the properties of the Fourier and Legendre series are different since at points x (−1, 1), the orthonormal Legendre polynomials increase infinitely.

If a function f(x) on the interval (−1, 1) is continuous and satisfies a Lipschitz condition, then the Fourier and Legendre series converges to a function uniformly over the interval (−1, 1).

The expansion in terms of a complete orthogonal system of adjoint Legendre functions (x), , …, (x) also plays an important part in problems of radiation heat transfer.

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