The eigenequation from the geometric theory of unstable resonators with nonuniform reflectivity outcouplers is solved by a series method. Either the reflectivity profile or the dominant mode can be specified and the other can be obtained. A spatially limited output can also be specified, but the resulting mode and reflectivity profile are not guaranteed to be spatially limited. This problem can be avoided, but a good general method for obtaining all of the desired features has not been found.