The integration path in Eq. (5) going around the two poles at and . In the limit , the contribution to the integral (10) of the second pole is half the residue of the integrand [see Eq. (11)].
(Color online) Solid line: The analytical -dispersion relation, for evaluated by using our results [Eqs. (20) and (21)]. Dashed line: The unphysical singularity reported in many previous works appears at when one considers instead the frequency obtained from Eq. (22). Stars: The direct calculation of the dispersion relation, via the numerical evaluation of the Landau integral (9). The numerical results are in perfect agreement with the analytical result, and show no sign of a singularity at .
Semilogarithmic plot of the initial distribution function , for (solid line). The dashed line represents a Maxwellian function in velocities.
(Top) Spectral electric energy as a function of the frequency for . (Bottom) Time evolution of the electric field spectral component (mode ) in semilogarithmic plot.
Time evolution of the electric field spectral component (mode ) in semilogarithmic plot, for a Maxwellian equilibrium distribution.
In both plots the dashed line represents the analytical solution while the stars represent the numerical simulation. (Top) Oscillation frequency vs wave number . (Bottom) Damping rate (absolute value) vs wave number .
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