A linear eigenfunction analysis of the beam–plasma instability for an annular beam interacting with a plasma filling the space between the conductors of a coaxial transmission line is presented. The beam and plasma electrons are modeled by relativistic cold fluids and the dispersion relation is three‐dimensional and fully electromagnetic. Particular attention is given to the radial eigenfunction structure and extensive numerical examples are presented. The behavior of the dispersion relation with various parameters is interpreted in terms of the infinite homogeneous theory. Simple analytic approximations for the dispersion relation for annular beams are given. Also, particle‐in‐cell simulation results are compared with the predictions of the linear analysis.