Exact complex source representations of time-harmonic radiation

New exact representations are presented for arbitrary compact source regions using distributions of sources in complex space. The equivalent sources are located on the complex ''sphere'' r = ia, with a weighting uniquely defined by the given far-field data. Each complex point source acts like a Gaussian beam with a well-defined directionality. This offers an exact wave-field representation suitable for arbitrary radiating fields, and optimal if the source is highly directional. In the limit of large ka the far-field pattern function is proportional to the associated weighting function for the complex sources. The theory is developed for both 2D and 3D, and numerical examples are presented for a 3D end-fire array.