[en] The theory of pair-wise coupled modes excited at a real frequency ω in a weakly nonuniform plasma (coordinate x) is developed on the basis of general local dispersion relations D(k,z) = 0 (such as from the Vlasov equation), which define a many-valued mapping of the real axis x = Rez onto the complex plane of the wavenumber k. Mode coupling, by definition, is the analytical continuation of the branch of the mapping and only occurs at the branch points. This requires that the z plane be cut along contours C/sub b/ given by D(k/sub c/,C/sub b/) = 0, partialD(k/sub c/,z)/partialk = 0, where z traces a contour passing through the branch points. The coupled modes can be analyzed by expanding the dispersion relation to second order in k around the saddle points and along the lines k/sub c/(x), yielding a system of dispersion relations corresponding to unambiguous second-order differential equations, guaranteed to have the right turning point behavior