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An improved method for the analysis of the fields within a section of nonuniform cylindrical waveguide is presented. The technique uses the method of successive approximations to obtain consistent solutions to the Stratton-Chu equations and the cylindrical harmonic representations of the fields that satisfy the Maxwell equations and the boundary conditions of the nonuniform waveguide. As opposed to some recently reported methods, this new method is not restricted to oversized waveguide structures and can directly evaluate the fields within a tapered section of waveguide. The method is validated by evaluating the fields within two different mode converters for gyrotron launchers. The accuracy is seen to be competitive with the commercial method of moments code, Surf3d, and requires a fraction of the computational resources.