Application of Algebraic Function Theory to Backward Wave Problems

Advancement of Algebraic Function Approximation in Eigenvalue Problems of Lossless Metallic Waveguides to Infinite Dimensions, Part Ii: Transfer of Results in Finite Dimensions to Infinite Dimensions

“…Similarly J 2 fin will mate with Y (p)Z(p) of again [1,2]. Hence we have established a correspondence between a finite system of exact eigenvalues and their finite dimensional eigenvectors corresponding to J 1 fin (or J 2 fin ) and a finite system of approximating eigenvalues and their finite dimensional eigenvectors corresponding to…”

“…Since as noted above for both J 1 fin and Z(p)Y (p), the characteristic equations are algebraic, without hesitation we can carry over the results obtained in the investigation of the eigenvalues of Z(p)Y (p) that were reported in [1,2], to the case of matrix J 1 fin . However it is to be noted that the roots of characteristic equation of J 1 fin , namely (1), are now exact eigenvalues of the infinite system.…”

“…Formulation of the problem as the solution of an algebraic equation was in fact the basis of the algebraic function approximation in [1,2]. But now in this way we have the advantage of containing the exact eigenvalues in an algebraic equation, for which there exists a well developed theory along with already obtained results of [1,2] for the assumed type of lossless closed guiding systems prescribed in Part I. [7, p. 368].…”

“…Note that the sets of generalized and proper eigenvectors ofẐ(p)Ŷ (p)andŶ (p)Ẑ(p), which we denoted by {v i } and {î i } respectively in Section 2 of Part I, are also bases in X and Y, to which {v Ji } and {î Ji } are alternatives. (1) of Part I, and were used in algebraic function approximation of the exact eigenvalues of the physical problem in [1,2].…”

“…In Section 3 we shall transfer the results on eigenvector inner products in finite dimensions obtained in [1][2][3][4], to infinite dimensions. We need to consider arguments of Section 3, in generalizing, in Section 4, results of algebraic function approximation in eigenvalue problems of lossless closed guides to exact solutions obtained in infinite dimensions.…”