 That the polynomials are complete means the following.Given any piecewise continuous function formula_46 with finitely many discontinuities in the interval [−1,1], the sequence of sums

converges in the mean to formula_46 as formula_49, provided we take

This completeness property underlies all the expansions discussed in this article, and is often stated in the form

with and .An especially compact expression for the Legendre polynomials is given by Rodrigues' formula