It is shown that the Floquet factor ${e}^{\mathrm{ik}(E)a}$ is analytic in the upper half complex energy plane, thus enabling a set of four dispersion relations to be derived from this expression as a direct result of the application of Cauchy's theorem. These relations are characterized by their ability to relate the wave number $k$ at one energy to the wave number at all others. In particular, the imaginary part of the wave number ${k}_{i}$ in the forbidden gap may be equated to an integral of a function of the real part of the wave number ${k}_{r}$ over allowed energies. As an application of these dispersion relations a theorem regarding the location of the branch points has been established.