Editing Legendre polynomials (section)

=== Definition by construction as an orthogonal system ===
In this approach, the polynomials are defined as an orthogonal system with respect to the weight function <math>w(x) = 1</math> over the interval <math> [-1,1]</math>. That is, <math>P_n(x)</math> is a polynomial of degree <math>n</math>, such that
<math display="block">\int_{-1}^1 P_m(x) P_n(x) \,dx = 0 \quad \text{if } n \ne m.</math>

With the additional standardization condition <math>P_n(1) = 1</math>, all the polynomials can be uniquely determined. We then start the construction process: <math>P_0(x) = 1</math> is the only correctly standardized polynomial of degree 0. <math>P_1(x)</math> must be orthogonal to <math>P_0</math>, leading to <math>P_1(x) = x</math>, and <math>P_2(x)</math> is determined by demanding orthogonality to <math>P_0</math> and <math>P_1</math>, and so on. <math>P_n</math> is fixed by demanding orthogonality to all <math>P_m</math> with <math> m < n </math>. This gives <math> n </math> conditions, which, along with the standardization <math> P_n(1) = 1</math> fixes all <math> n+1</math> coefficients in <math> P_n(x)</math>. With work, all the coefficients of every polynomial can be systematically determined, leading to the explicit representation in powers of <math>x</math> given below.

This definition of the <math>P_n</math>'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, <math> x, x^2, x^3, \ldots</math>. Finally, by defining them via orthogonality with respect to the [[Lebesgue measure]] on <math> [-1, 1] </math>, it sets up the Legendre polynomials as one of the three [[classical orthogonal polynomials|classical orthogonal polynomial systems]]. The other two are the [[Laguerre polynomials]], which are orthogonal over the half line <math>[0,\infty)</math> with the weight <math> e^{-x} </math>, and the [[Hermite polynomials]], orthogonal over the full line <math>(-\infty,\infty)</math> with weight <math> e^{-x^2} </math>.