Legendre polynomials  In physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 1782) are a system of complete and orthogonal polynomials, with a vast number of mathematical properties, and numerous applications. They can be defined in many ways, and the various definitions highlight different aspects as well as suggest generalizations and connections to different mathematical structures and physical and numerical applications. Closely related to the Legendre polynomials are associated Legendre polynomials, Legendre functions, Legendre functions of the second kind, and associated Legendre functions. In this approach, the polynomials are defined as an orthogonal system with respect to the weight function formula_1 over the interval formula_2. That is, formula_3 is a polynomial of degree formula_4, such that  This determines the polynomials completely up to an overall scale factor, which is fixed by the standardization formula_6. That this is a constructive definition is seen thus: formula_7 is the only correctly standardized polynomial of degree 0. formula_8 must be orthogonal to formula_9, leading to formula_10, and formula_11 is determined by demanding orthogonality to formula_9 and formula_13, and so on. formula_14 is fixed by demanding orthogonality to all formula_15 with formula_16. This gives formula_17 conditions, which, along with the standardization formula_18 fixes all formula_19 coefficients in formula_20. With work, all the coefficients of every polynomial can be systematically determined, leading to the explicit representation in powers of formula_21 given below. This definition of the formula_14's is the simplest one. It does not appeal to the theory of differential equations. Second, the completeness of the polynomials follows immediately from the completeness of the powers 1, formula_23. Finally, by defining them via orthogonality with respect to the most obvious weight function on a finite interval, it sets up the Legendre polynomials as one of the three classical orthogonal polynomial systems. The other two are the Laguerre polynomials, which are orthogonal over the half line formula_24, and the Hermite polynomials, orthogonal over the full line formula_25, with weight functions that are the most natural analytic functions that ensure convergence of all integrals.