This thesis is concerned with the results of the application of circle polynomials to two main processes involving the use of the scalar diffraction integral over a circular region. The choice of a preferred description of the far-field radiation pattern from a circular aperture antenna, is shown to give rise to the use of circle polynomials in the description of the aperture radial amplitude function. A duality then exists whereby both functions are described by series with essentially the same coefficients. This enables the two processes, of analysing the patterns from prescribed aperture distributions and of synthesising prescribed patterns and determining the required aperture distribution, to be developed from the same mathematical procedure. Several examples, drawn from practical antenna studies, are used to illustrate the analytical method and the radiation patterns are given in closed form. The reverse process turns out to be very limited in the possible range of antenna patterns that can be synthesised. This is due to the natural limitations of radiation pattern shaping by amplitude methods alone and agrees with the limitations of the mathematical process. The most advantageous distributions found by this method have discontinuous amplitude steps. Final work is therefore done on patterns constructed from stepped aperture distributions such as would occur naturally in an antenna constructed of an array of rings or zones.