Substituting equation (5.49) for the covariance kernel, and assuming that

and

the solution of equation (5.48) reduces to the solution of the equation

where

The solution of equation (5.52) is the product of the individual solutions of the
two equations

and

Differentiating each of equations (5.54) and (5.55) twice, two second order
differential equations are obtained along with their associated boundary
conditions. The solution of the first of these equations produces
the following eigenvalues and normalized eigenfunctions ,

and

In equations (5.56)-(5.58), the symbol refers to the solution of the following transcendental equation

The solution of equation (5.55) is identical to equations (5.56)-(5.59), with
replaced by . Note that if , then to each eigenvalue there correspond two eigenfunctions
of the form given by equation (5.50). The second function being obtained from
the first one by permuting the subscripts. In this case, therefore, the complete normalized
eigenfunctions are given by the equation

In the expansion of the random process, the terms are ordered in descending order of the magnitude of the eigenvalues .

Sat Oct 28 13:18:26 EDT 1995