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We present an analytical method, employing toroidal harmonics, for computing the three-dimensional (3-D) magnetic field from a circular cylindrical bipolar permanent magnet. Bipolar magnets are those which are polarized perpendicular to the axis of the cylinder. We take a completely analytical approach in order to facilitate parametric studies of the external 3-D magnetic field produced by bipolar magnets. The results of our analysis are verified by comparing them to previously published results. The application of toroidal harmonics are ultimately shown to be well-suited for both parametric studies as well as numerical computation.