The new capabilities of AC calorimetry, when working at frequencies above the classical limit, were shown. The idea of the method was to use the information about the phases and the amplitudes of the temperature oscillations on both sides of a platelike sample for simultaneous determination of the sample's complex heat capacity C-s(omega) and thermal conductivity kappa(s)(omega), when the oscillating heat flow was supplied to one face of the sample. The mathematical algorithm for simultaneous determination of real and imaginary parts of the sample's heat capacity and thermal conductivity was developed. The advanced AC technique was applied for polymers. As a first step, the approximation of the frequency independent thermal conductivity was used to determine the complex heat capacity C-s(omega) of polymers in the melting region. It was found that the processes of reversing (more definitely heat-reversing) melting and completely irreversing melting can be distinguished by the complex heat capacity measurements. Thus, the different physical processes at melting can be distinguished by the complex heat capacity measurements.