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The analytical solution for the eddy-current induction in a conducting half-space containing an infinitely long cylindrical hole parallel to the interface is derived by means of the second-order vector potential and the domain truncation technique. The eddy currents are induced by a cylindrical coil which can be placed either in the nonconductive region above the half-space or inside the hole or in both locations simultaneously. Analytical expressions for the coil impedance are derived in both cases. The problem is treated as a coupled scattering problem and the derived analytical expressions permit a clear physical interpretation of the different terms, thus enhancing the understanding of underlying physics. The validity of the solution is demonstrated by comparisons to results obtained with the finite element method.