A formalism for absolute and convective instabilities in parallel shear flows is extended to the three-dimensional case. Assuming that the dispersion relation function is given by D(k, l, omega), where k and l are wave numbers, and omega is a frequency, the analytic criterion is formulated by which a point (k0, l0, omega-0) with Im omega-0 > 0 contributes to the absolute instability if and only if one of the two equivalent conditions is satisfied: (i) At least two roots in l of the system D(k, l, omega) = 0, D(k)(k, l, omega) = 0, originating on opposite sides of the real l-axis, collide on the l-plane for the parameter values k0, l0, omega-0, as omega is brought down to omega-0. Every point on the k-plane, that corresponds to a point on the collision paths on the l-plane, is itself a coalescence point of k-roots for a fixed l of D(k, l, omega) = 0, that originate on opposite sides of the real k-axis. (ii) At least two roots in k of the system D(k, l, omega) = 0, D(l)(k, l, omega) = 0, originating on opposite sides of the real k-axis, collide on the k-plane for the parameter values k0, l0, omega-0, as omega is brought down to omega-0. Every point on the l-plane, that corresponds to a point on the collision paths on the k-plane, is itself a coalescence point of l-roots for a fixed k of D(k, l, omega) = 0, that originate on opposite sides of the real l-axis. Consequently, the causality condition for spatially amplifying 3-D waves in absolutely stable, but convectively unstable flow is derived as follows. We denote by (alpha, beta) a unit vector on the (x, y) plane. The contributions to amplification in the direction of this vector come from the end points of the trajectories that consist of the coalescence roots on the l1-plane, given by l1 = -beta-k + alpha-l, of the system D = 0, -beta-D(k) + alpha-D(l) = 0. The k1-components of these trajectories have to pass from above to below the real axis on a k1-plane, given by k1 = alpha-k + beta-l, as omega moves down to omega-0. Here omega-0 is the real frequency of excitation. At each point of such trajectories the group velocity vector (D(k), D(l)) is collinear with the direction vector (alpha, beta). There exists a direction for which the spatial amplification rate reaches its maximum. The formalism is illustrated with a simple model example. A procedure for computing the N-factor in the e(N)-method, which is based on the wave packet approach is developed.
