Of the many novel coronagraphic and nulling techniques that have been suggested to improve image contrast for exoplanet detection, one of the most promising is the Quadrant Phase Mask suggested by Rouan et al. Analysis of this optical system has previously been performed by discrete Fourier transform methods, that result in systematic errors due to the implicit assumptions of the methods and mathematical singularities in the transform of the phase mask. In this paper, we describe an analytical treatment of this optical system that treats these singularities explicitly. We calculate the leakage of a Quadrant Phase Mask Coronagraph with these analytical techniques, and show that a Quadrant Phase Mask rejects all on-axis light for an unaberrated, unobscured circular aperture and is therefore a nearly perfect coronagraph. We demonstrate why the Quadrant Phase Mask coronagraph suffers degraded performance with an obscured aperture, and propose modifications to the pupil geometry to mitigate this problem.