A positive slope in a velocity distribution function perpendicular to the ambient magnetic field, such as due to a loss cone or ring velocity distribution, can become a free energy source for the excitation of various plasma waves. Since there exists no analytic expression for integrals of Maxwellian ring velocity distribution functions, their linear properties have previously been studied using several approximations or modeled distributions. In this paper, a numerical method for analyzing the linear dispersion relation for Maxwellian ring-beam velocity distributions is developed. The obtained linear properties are confirmed by direct comparison with full particle simulation results.
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