The theory of pair‐wise coupled modes excited at a real frequency &ohgr; in a weakly nonuniform plasma (coordinatex) is developed on the basis of general local dispersion relationsD(k,z) = 0 (such as from the Vlasov equation), which define a many‐valued mapping of the real axisx= Rezonto the complex plane of the wavenumberk. Mode coupling, by definition, is the analytical continuation of the branch of the mapping and only occurs at the branch points. This requires that thezplane be cut along contoursCbgiven byD(kc,Cb) = 0, ∂D(kc,z)/∂k= 0, whereztraces a contour passing through the branch points. The coupled modes can be analyzed by expanding the dispersion relation to second order inkaround the saddle points and along the lineskc(x), yielding a system of dispersion relations corresponding to unambiguous second‐order differential equations, guaranteed to have the right turning point behavior.