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The problems of the diffraction of scalar waves by a circular aperture in a perfectly soft, and in a perfectly rigid, infinite, planar screen are treated. New integral representations of the solution are presented which have the virtue of automatically satisfying the time-reduced wave equation, the radiation condition and the boundary conditions. The unknown functions appearing in these representations are shown to satisfy Fredholm-type integral equations of the second kind which yield, after iteration, accurate, approximate solutions when (the product of the wave number and the aperture radius) is sufficiently small. These approximate solutions are in turn employed to calculate the aperture fields, the far fields, the transmission coefficients, and the edge behaviors. The results are in complete agreement with known results and for some of these quantities they are more accurate in that higher powers of are included. Finally, in the case of the rigid screen problem, for all values of , we determine the form of the edge behavior and give a simple proof of the unique existence of the solution.