A Fourier representation of the electric field volume integral equation for an open and cylindrically symmetrical waveguide is given in this paper. The waveguide material is assumed to be isotropic, non-magnetic and with an arbitrary radial variation in the relative permittivity. The Fourier representation yields a system of one-dimensional integral equations, one system for each azimuthal index and where the Fourier variable for the longitudinal direction plays the role of a spectral parameter. The integral equation is of the second kind and has a kernel that is generally discontinuous on the diagonal and singular at the origin. In the axial symmetric case, it can readily be shown that the elements of the matrix kernel belong to an L2-space, and hence that the integral operator is compact and the analytic Fredholm theorem is applicable.