Substituting equation (5.49) for the covariance kernel, and assuming that
the solution of equation (5.48) reduces to the solution of the equation
The solution of equation (5.52) is the product of the individual solutions of the two equations
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 ,
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 .