A functional analytic approach to the linearized collisionless Vlasov equation is presented utilizing a resolvent integration technique on the resolvent of the transport operator evaluated at a particular point. Formulae for the eigenfunction expansion are found for cases in which the plasma disperion function &Lgr; has first and second order zeroes. Special care is taken in the study of real zeroes of &Lgr; culminating in new results for this case. For a simple zero of &Lgr; with nonvanishing imaginary part the van Kampen–Case discrete modes are reproduced. The results are used to obtain the solution to the initial value problem.