The comparison equation method is used to study the outer expansions of the solutions of the Orr-Sommerfeld equation. All but one of these expansions are multiple-valued and must therefore exhibit the Stokes phenomenon. One of the major aims of the present paper is to obtain first approximations to the Stokes multipliers which describe the continuation of these expansions on crossing a Stokes line in the complex plane. By restricting the domains of validity of these expansions appropriately we can insure that all of the expansions are `complete' in the sense of Olver and this is an essential feature of the work. The resulting approximations show that, in some sectors, a sharp distinction can no longer be made between approximations of inviscid and viscous type. A consistent first-order approximation to the characteristic equation in the complete sense is derived and compared with the more usual second-order approximation of Poincare type. Calculations of the curve of neutral stability for plane Poiseuille flow clearly show that a first approximation in the complete sense provides a substantially better approximation to the neutral curve than a second approximation in the Poincare sense.