In the present paper we construct analytical solution of eddy current problems in cylindrical coordinates where a coil with alternating current is located inside or outside a multilayer tube. It is assumed that the electric conductivity and magnetic permeability of conducting layers of the tube are power functions of radial coordinate. The problem is solved by the method of the Fourier transform. The system of ordinary differential equations for the vector potential in te transformed space can be solved in terms of different special functions. Results of numerical calculations are presented.