In this paper is presented an analytical study of the azimuthal phase-mask coronagraph currently envisioned for detecting and characterizing extra-solar planets. Special emphasis is put on the physical and geometrical interpretation of the mathematical development. Two necessary conditions are defined for achieving full extinction in the pupil plane of the coronagraph, stating that the complex amplitude generated by the phase mask should have zero average, on the one hand, and its Fourier coefficients should only be even, on the other hand. Examples of such phase functions are reviewed, including optical vortices, four-quadrant phase masks, and azimuthal cosine phase functions. Hints for building more sophisticated functions are also given. Finally, a simplified expression of light leaks due to mask imperfection is proposed
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