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It is demonstrated by specific examples that eigenvalues of the electric field integral equation (EFIE) operator for finite extent objects in lossless media can have branch points in the complex frequency -plane. Specifically, it is shown from the analysis of the spheroidal wave equation in the -plane that branch points are present in the eigenvalues of the integral operator associated with the scalar scattering problem from a perfectly conducting spheroid. Similarly, it is concluded from the analysis of the Mathieu's differential equation that branch points, besides that associated with the infinite extent of the object, exist in the -plane behavior of the eigenvalues of the EFIE operator for a perfectly conducting infinite elliptic cylinder. A proof is given that branch point singularities cannot occur in the eigenvalues of EFIE operators for structures in which geometrical symmetry completely determines the eigenfunctions (and hence they are frequency independent). It is conjectured that branch points may always be present when sufficient object symmetry is lacking. This conjecture is supported by the fact that branch points appear when a sphere is deformed into a spheroid or when a circular cylinder is deformed into an elliptic one. An analogous phenomenon has been observed in circuit and transmission line problems. For example, it is shown that when the taper parameter of an exponential transmission line goes to zero (the line becomes uniform and thus a symmetrical structure), the branch points in the eigenvalues of the impedance matrix move away to infinity.