ABSTRACT
We pass now from the description of the properties of individual eigenvalues and eigenfunctions to the principal property enjoyed by the eigenfunctions as a set. This is represented by the theory of “eigenfunction expansions”; an “arbitrary” function, of the several variables involved, can be expanded as a series of eigenfunctions, products of solutions of the individual ordinary differential equations that make up the eigenvalue problem. Since the eigenfunctions are mutually orthogonal, with respect to a determinantal weight-function, the coefficients in such an expansion are easily found, given the function to be expanded. The difficulty is, of course, to show that the function being “expanded” is indeed represented by the series of eigenfunctions, even when the latter series is known to be convergent.
