A normal mode treatment is proposed for the solution of boundary value problems in plasma oscillations. The plasma distribution function and electric field are expanded in terms of the solutions of the coupled Vlasov and Maxwell equations. These normal mode solutions are somewhat unusual in that they are singular functions of the velocity variable, and, in fact, the expansion theorem is proved by means of a direct solution of an associated singular integral equation. As an application of the method we obtain the impedance of a plasma-filled parallel plate condenser.