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The Mattis-Bardeen theory for the anomalous skin effect in the superconductors has been extended taking the complex gap energy into account and the surface impedance of superconductors has been calculated using the extended Mattis-Bardeen theory. It is found that the surface resistance of the superconductor increases with increasing magnitude of the imaginary part of the gap energy. It is demonstrated that the calculated surface resistance for a NbN film quantitatively agrees with the measured one. Itis also found that temperature dependence of Q values of super conducting resonators is well describedby those calculated by the extended Mattis-Bardeen equation

RF Conductivity and Surface Impedance of a Superconductor Taking into Account the Complex Superconducting gap Energy