  Existing analytical models for transverse beam dynamics in isochronous
cyclotrons are often not valid or not precise for relativistic energies. The
main difficulty in developing such models lies in the fact that cross-terms
between derivatives of the average magnetic field and the azimuthally varying
components cannot be neglected at higher energies. Taking such cross-terms
rigorously into account results in an even larger number of terms that need to
be included in the equations. In this paper, a method is developed which is
relativistic correct and which provides results that are practical and easy to
use. We derive new formulas, graphs and tables for the radial and vertical
tunes in terms of the the flutter, its radial derivatives, the spiral angle and
the relativistic gamma. Using this method, we study the $2\nu_r=N$ structural
resonance ($N$ is number of sectors) and provide formulas and graphs for its
stopband. Combining those equations with the new equation for the verical tune,
we find the stability zone and the energy limit of compact isochronous
cyclotrons for any value of N. We confront the new analytical method with
closed orbit simulations of the IBA C400 cyclotron for hadron therapy.
