Two types of 3D boundary value problems are solved for boundary conditions on a closed toroidal surface with axial or N-fold symmetry around the Z axis of a cylindrical coordinate system R, Φ, Z. In case I a surface current distribution is computed which generates inside the torus a given magnetic vacuum field. In case II a vacuum field within the torus with finite toroidal flux and zero surface normal is calculated. The method of solution is based on the availability of some complete set of auxiliary field harmonics which are regular outside the torus in case I and regular inside the torus in case II. The applied formalism is presented together with a new complete set of special field harmonics suited for case II and especially for the analytical representation of stellarator fields. The convergence of the solution method is tested and examples relating to stellarator field configurations are shown
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W. Dommaschk (1981) studied this question.