 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.