Appl. sci. Res. Section B, Vol. 8 ON THE SOLUTION OF AN EIGENVALUE EQUATION OF THE WIENER-HOPF TYPE IN FINITE AND INFINITE RANGES *) by R. MITTRA Antenna Laboratory, Department of Electrical Engineering, University of Illinois, Urbana, Illinois, U.S.A. Summary The purpose of the paper is to present a formulation of the eigenvalue matrix equation of the Wiener-Hopf integral equation defined in finite and infinite ranges. The method provides a simple means for obtaining the eigen- value equation and indicates a way for obtaining the eigenfunctions and the eigenvalue. The important contribution of the paper is the direct rather than the transform method of solution. Such a formulation is also helpful in the solution of inhomogeneous Wiener-Hopf equations in finite and infinite ranges. w 1. Introduction. In this paper we consider the eigenvalue problem defined by the equation ~T(t) = /, q~(-r) K It - ~-I d~-, 0 < T < co, (1.1) K ]tl = E Cr exp(-- kr Itl), (i.2) r=l which is encountered in a class of problems in probability theory. The origin of (1.1) and a transform method of solution has been discussed by Youla 1) in a recent paper. The purpose of the paper is to show that the eigenvalue problem can be formulated in terms of a determinantal equation without resorting to the transform method of formulation. The direct method of formulation is simple and involves comparatively less amount of work. From the assumed form of the kernel it is seen that the method is limited to the *) The research reported in this paper was carried out under Contract No. AF 33 (616)-6079 at the University of Illinois. - - 201 - -