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Magnetic Field Expansion Out of a Plane: Application to Inverse Cyclotron Muon Cooling

In studies of the dynamics of charged particles in a cyclotron magnetic field, the specified field is generally B_z in the z = 0 midplane where B_r and B_theta are zero. B_r(r, theta, z) and B_theta(r, theta, z) are simple to determine through a linear expansion which assumes that B_z is independent of z. But, an expansion to only first order may not be sufficient for orbit simulations at large z. This paper reviews the expansion of a specified B_z(r, theta, z = 0) out of the z = 0 midplane to arbitrary order, and shows simple examples worked out to 4th order.