Bremmer series that correct parabolic approximations

Abstract A Bremmer type series solution of the three dimensional reduced wave equation is obtained. The series is obtained by iterating generalizations of the Bellman-Kalaba integral equations. The lowest order term is the solution of the parabolic approximation to the reduced wave equation. The series thus provides systematic corrections to the parabolic approximation. New derivations of the parabolic approximation are also provided. These are based on the idea of splitting a solution to the reduced wave equation into “upward” and “downward” components.