  We propose a new and effective method to find plasma oscillatory and wave
modes. It implies searching a pair of poles of two-dimensional (in coordinate
$x$ and time $t$) Laplace transform of self-consistent plasma electric field
$E(x,t) \to E_{p_1p_2}$, where $p_1 \equiv -i \omega$, $p_2 \equiv i k$ are
Laplace transform parameters, that is determining a pair of zeros of the
following equation $$\frac1{E_{p_1p_2}} = 0 .$$ This kind of conditional
equation for searching double poles of $E_{p_1p_2}$ we call ``general
dispersion equation'', so far as it is used to find the pair values
($\omega^{(n)}, k^{(n)}$), $n=1, 2, ...$ . It differs basically from the
classic dispersion equation $\epsilon_l(\omega,k) = 0$ (and is not its
generalization), where $\epsilon_l$ is longitudinal dielectric susceptibility,
its analytical formula being derived according to Landau analytical
continuation. In distinction to $\epsilon_l$, which is completely plasma
characteristic, the function $E_{p_1p_2}$ is defined by initial and boundary
conditions and allows one to find all the variety of asymptotical plasma modes
for each concrete plasma problem. In this paper we demonstrate some
possibilities of applying this method to the simplest cases of collisionless
ion-electron plasma and to electron plasma with collisions described by a
collision-relaxation term $-\nu f^{(1)}$.
