Analytic solutions to the problem of linear wave conversion in a cold, unmagnetized, collisionless plasma are obtained for nearly all angles of incidence. The resulting absorption coefficients are in excellent agreement with numerical work in the appropriate limits and differences in previous solutions are reconciled. The warm plasma equations are also solved, and the independence of the absorption coefficient to thermal effects verified. The singularities neglected by Piliya are properly treated by viewing the frequency dependence as arising from a Laplace transform.