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If n is an odd integer, the second solution terminates after a finite number of terms, while the first solution produces an infinite series. The finite solutions are the Legendre Polynomials, also known as solutions of the first kind, denoted by P n ( x ) . n denoted by Q ( x ) . ) ( 2 n − 2 k ) ! ( n − k ) ! ( n − 2 k ) ! − 4! 6! B x − 3! 3!
The Legendre polynomials, sometimes called Legendre functions of the first kind, Legendre coefficients, or zonal harmonics (Whittaker and Watson 1990, p. 302), are solutions to the Legendre differential equation. If l is an integer, they are polynomials. The Legendre polynomials P_n(x) are illustrated above for x in [-1,1] and n=1, 2, ..., 5.
24 Μαΐ 2024 · Legendre Polynomials are one of a set of classical orthogonal polynomials. These polynomials satisfy a second-order linear differential equation. This differential equation occurs naturally in the solution of initial boundary value problems in three dimensions which possess some spherical symmetry.
LEGENDRE POLYNOMIALS - RODRIGUES FORMULA AND ORTHOGONALITY 3 Since m<n, the derivative inside the integral is zero, since the largest power of xin (x2 1)m is x2m and 2m<m+n. Therefore, the over-all integral is zero, and we have shown that the Legendre polynomials are orthogonal (that is, 8 is true). What if n= m?
The following lecture introduces the Legendre polynomials. It includes their derivation, and the topics of orthogonality, normalization, and recursion. We start with a solution to the Laplace equation in 3 - dimensional space: This is Legendre's Equation. k! Hn-kL! Hn-2 kL!
24 Μαΐ 2024 · These polynomial solutions are the Legendre polynomials, which we designate as \(y(x)=P_{n}(x)\). Furthermore, for \(n\) an even integer, \(P_{n}(x)\) is an even function and for \(n\) an odd integer, \(P_{n}(x)\) is an odd function.
Legendre polynomials occur in the solution of Laplace's equation of the static potential, ∇ 2 Φ(x) = 0, in a charge-free region of space, using the method of separation of variables, where the boundary conditions have axial symmetry (no dependence on an azimuthal angle).
