We derive the Debye-series expansion of the partial-wave scattering and interior amplitudes for the interaction of a diagonally incident beam of arbitrary profile with an infinitely long homogeneous dielectric circular cylinder. We then discuss the physical interpretation of the various terms of the series. We also consider the one-internal-reflection Debye-series terms for diagonal plane-wave incidence and examine the first-order rainbow extinction transition as a function of the tilt angle of the incident plane wave. We experimentally observe the first-order rainbow extinction transition and compare our observations with the Debye-series predictions. (C) 1997 Optical Society of America.