In this paper we analyze characteristics of the dispersion function for leaky-wave modes in the vicinity of cutoff for several representative waveguiding structures. Our principal purpose is to demonstrate that in the vicinity of leaky-wave cutoff in open-boundary waveguides (in the spectral-gap region), dispersion behavior is controlled by the presence of branch points in the complex frequency plane. A similar situation occurs for the ordinary modes of homogeneously filled, perfectly conducting cylindrical waveguides. These closed waveguides admit to simple analysis, leading to an explicit dispersion function which indicates frequency domain branch points. For open-boundary waveguides, the presence of frequency domain branch points is obscured by the necessity of numerically solving an implicit dispersion equation. A set of sufficient conditions is provided here which defines these branch points in a unified manner for both open and closed waveguides. Identification of these points allows for rapid determination of important and interesting regions in both the frequency and wavenumber planes and leads to increased understanding of dispersion behavior, especially in the case of dielectric loss. Examples are shown for several waveguiding geometries to demonstrate the general nature of the presented formulation. ¿ 1998 American Geophysical Union