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tokens: that an analytic continuation of B ( c ) to the lower half-plane c , apart from poles of Landau quasi-modes, contains also a pole due to the damped mode c = c − , and near this pole B ( c ) has the form: where For real c , expression ( 23 ) can be decomposed, with the aid of equation ( 21 ), as follows: where Now analytic continuation is obvious, since ε ± ( c ) can be replaced by off the real axis, and g ± are replaced by where, depending on signs ‘’ or ‘’, integration contour passes below or above the singularity at u = c . Note that i.e. this is a decomposition of function g (). Since has one zero in the lower half-plane c = c − , B − ( c ) has a pole at this point leading to ∝exp (− γ k t ) contribution to density decay. Expansion near c = c − gives Finally, substitution to leads to the desired result ( 35 ), if we choose the integration contour along the real u -axis. The input from this pole gives a density decay slower that the Landau damping for k > k * ≈ 0.437, as is seen in Fig. 8 . Note that decomposition ( 37 ) is not unique. Moreover, it is possible to decompose B ( c ) so that parts of new decomposition have no singularities in the upper/lower half-planes (Appendix B ). Gaussian in time density decay occurred for Maxwell B ( c ) (Fig. 6 ) is simply a reflection of the fact that B ( c ) has no singularities in the lower half-plane. 4.3 Power-law decay To understand an appearance of the power-law term (Fig. 7 ), we should re-examine derivation of equations ( 7 ) and ( 8 ). They are obtained as a result of analytic continuation of the integrals over the real u -axis by deformation of the integration contour to the lower half-plane. Since g () is zero for || > * , the cut in the complex plane is finite (Fig. 10 ), and deformation of the contour is not needed for this integral. On the either side of the cut The cut leads to an additional input to density ( 6 ): Integrating by parts, assuming g () is even and real, one obtains for large t : Expression in the square brackets gives sinusoidal oscillations with some phase shift due to the presence of in the denominator. If g ( * ) is small, the power-law decay reveals itself only after a while. Figure 10. Integration contour in ( 6 ) in the complex -plane (blue dashes). ‘+’ signs mark the Landau-damped quasi-modes obtained as zeros of . A wavy line marks the cut on real axis c : − * c * . 5 IMPLICATIONS In this article, we analysed numerical solutions of equations describing perturbations of the homogeneous stellar medium. Although the equilibrium state is problematic in the infinite media and invokes a so-called Jeans swindle (e.g. BT ), it is a good testing ground for many phenomena in stellar dynamics (see Bisnovatyi-Kogan Zel’dovich 1971 , for example of a more realistic set-up). The unexpected behaviour of damped solutions in the unstable domain is associated with the presence of a genuine damped mode, which is complex conjugate to the unstable Jeans mode. The damped mode cannot