Thus, using the calculus of residues:  where the line integral is over a contour "C" that includes all the eigenvalues of "L". Suppose our functions are defined over some coordinates {"xj"}, that is:  Introducing the notation  where "δ(x − y)" = "δ(x1 − y1, x2 − y2, x3 − y3, ...)" is the Dirac delta function, we can write  Then:  The function "G(x, y; λ)" defined by:  is called the "Green's function" for operator "L", and satisfies:  Consider the operator equation:  in terms of coordinates:  A particular case is "λ" = 0. The Green's function of the previous section is:  and satisfies:  Using this Green's function property:  Then, multiplying both sides of this equation by "h"("z") and integrating:  which suggests the solution is:  That is, the function "ψ"("x") satisfying the operator equation is found if we can find the spectrum of "O", and construct "G", for example by using:  There are many other ways to find "G", of course. See the articles on Green's functions and on Fredholm integral equations. It must be kept in mind that the above mathematics is purely formal, and a rigorous treatment involves some pretty sophisticated mathematics, including a good background knowledge of functional analysis, Hilbert spaces, distributions and so forth. Consult these articles and the references for more detail. Optimization problems may be the most useful examples about the combinatorial significance of the eigenvalues and eigenvectors in symmetric matrices, especially for the Rayleigh quotient with respect to a matrix M.  Theorem "Let M be a symmetric matrix and let x be the non-zero vector that maximizes the Rayleigh quotient with respect to M. Then, x is an eigenvector of M with eigenvalue equal to the Rayleigh quotient. Moreover, this eigenvalue is the largest eigenvalue of M. "  Proof Assume the spectral theorem. Let the eigenvalues of M be formula_50. Since the {formula_51} form an orthonormal basis, any vector x can be expressed in this basis as  The way to prove this formula is pretty easy. Namely,  evaluate the Rayleigh quotient with respect to x:  where we used Parseval's identity in the last line.