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legendr
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There are 7 letters in LEGENDR ( D2E1G2L1N1R1 )
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LEGENDR - In mathematics, Legendre polynomials (named after Adrien-Marie Legendre) are the polynomial solutions P ...
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In mathematics, Legendre polynomials (named after Adrien-Marie Legendre) are the polynomial solutions * * * * * P * * n * * * ( * x * ) * * * {\displaystyle P_{n}(x)} * to Legendre's differential equation * ‹The template Repeat is being considered for deletion.› * ‹The template Repeat is being considered for deletion.› with integer parameter * * * * n * ≥ * 0 * * * {\displaystyle n\geq 0} * and with the convention * * * * * P * * n * * * ( * 1 * ) * = * 1 * * * {\displaystyle P_{n}(1)=1} * . The * * * * * P * * n * * * ( * x * ) * * * {\displaystyle P_{n}(x)} * form a polynomial sequence of orthogonal polynomials of degree n. They can be expressed using Rodrigues' formula: * * * * * * P * * n * * * ( * x * ) * = * * * 1 * * * 2 * * n * * * n * ! * * * * * * * * d * * * n * * * * * d * * * x * * n * * * * * * * * ( * * * x * * 2 * * * − * 1 * * ) * * * n * * * * , * * * {\displaystyle P_{n}(x)={\frac {1}{2^{n}n!}}{\frac {\mathrm {d} ^{n}}{\mathrm {d} x^{n}}}\left(x^{2}-1\right)^{n}\,,} * or any of the other representations given below. * The differential equation admits another, non-polynomial solution, the Legendre functions of the second kind * * * * * Q * * n * * * * * {\displaystyle Q_{n}} * , discussed below. * A two-parameter generalization of (Eq. 1) is called Legendre's general differential equation, solved by the Associated Legendre polynomials. Legendre functions are solutions of Legendre's differential equation (generalized or not) with non-integer parameters. * Legendre's 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 generating function is the basis for multipole expansions. |