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In eddy-current sensors and eddy-current testing, it is important to analyze the magnetic field of the coil. Toward this end, we analyzed magnetic fields in various coils, such as circular and rectangular coils. We used the double Fourier transformation and applied it to the eddy-current problem of coils facing a sheet conductor. However, only a limited number of coil shapes could be analyzed by this method. We therefore devised a method of magnetic field analysis that could be applied to a variety of coils by double Fourier transformation. We focused attention on the generated magnetic field of the coil, which is a basic characteristic. Our analysis method is made applicable to a much larger variety of coil shapes by defining a completely new type of function, called shape functions. We verified the validity of the analytical results experimentally. Although the Biot-Savart law may be used to analyze magnetic fields of coils, the law is not applicable when the coil faces a sheet conductor.