The dispersive properties of a plasma with a generalized Cauchy equilibrium velocity distribution of the form (v2+a2)−3are derived for certain modes of propagation. The plasma dispersion function is found in a simple algebraic form which makes the solution of the dispersion equations readily accessible. A quartic dispersion relation which is obtained for right‐hand circularly polarized electromagnetic waves propagating along a uniform static magnetic field is solved for all frequencies including cyclotron resonance. The complex index of refraction for this mode is plotted for several values of the temperature and the plasma‐cyclotron frequency ratio. In general, thermal effects reduce the infinite discontinuity in phase at the cyclotron resonance to a finite value and cause thermal damping of the propagating wave below resonance. The results are compared with previous numerical analyses using the Maxwell distribution.