A unified theory is formulated for the determination of the functional behaviour in the frequency domain of the propagation constant of electromagnetic waves which may exist in a uniform inhomogeneous waveguide which contains a linear time-invariant passive medium. By applying the constraints of linearity and time invariance to a bounded uniform medium, it is shown that the propagation coefficient in the transformed frequency domain is solely a function of the complex frequency variablep. The additional physical constraints of reality and passivity are shown to imply that the propagation constant γ belongs to a special class of functions which have been termed ‘nonzero real functions’. If the waveguide is also dissipationless, γ becomes a ‘symmetrical nonezero real function’ possessing symmetry about the axisp=jω. In restricted geometries for the cross-section of the guide, additional constraints on γ may exist and, as an example, a rectangular waveguide partially loaded with dielectric material is cited. A qualitative discussion on closed microstrip lines and circular waveguides containing dielectric rods is also included.