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We employ a toroidal harmonic expansion in order to develop a three-dimensional solution for the magnetic field due to a permanent-magnet multipole cylinder. The equations derived in this paper can be used for the optimization and design of various devices that employ cylindrical multipole magnets. The analytical equations employ hypergeometric functions derived from the analytical integration of zeroth-order toroidal functions. Hypergeometric functions are quite general and are very useful for parametric studies.