Verifications of the theory for extension of the algebraic function approximation in closed, lossless guiding system eigenvalue problems to infinite dimensions, are given. Numerical verifications are made on a closed circular guide loaded with a coaxial dielectric rod. The defective nature of an eigenvalue at the onset of a complex wave mode is demonstrated both in the algebraic function approximation and in infinite dimensions. At a cutoff frequency, for the exact infinite system, analyticity of the eigenvalue, but singularity of the propagation constant, are checked numerically. It is observed that in exact analysis too, by itself a complex wave mode does not take part in energy transportation, but a pair of complex conjugate modes behaves as an evanescent wave.