Thermal plasma of solar atmosphere includes a wide range of temperatures. This plasma is often quantified, both in observations and models, by a differential emission measure (DEM). DEM is a distribution of the thermal electron density square over temperature. In observations, the DEM is computed along a line of sight, while in the modeling -- over an elementary volume element (voxel). This description of the multi-thermal plasma is convenient and widely used in the analysis and modeling of extreme ultraviolet emission (EUV), which has an optically thin character. However, there is no corresponding treatment in the radio domain, where optical depth of emission can be large, more than one emission mechanism are involved, and plasma effects are important. Here, we extend the theory of the thermal gyroresonance and free-free radio emissions in the classical mono-temperature Maxwellian plasma to the case of a multi-temperature plasma. The free-free component is computed using the DEM and temperature-dependent ionization states of coronal ions, contributions from collisions of electrons with neutral atoms, exact Gaunt factor, and the magnetic field effect. For the gyroresonant component, another measure of the multi-temperature plasma is used which describes the distribution of the thermal electron density over temperature. We give representative examples demonstrating important changes in the emission intensity and polarization due to considered effects. The theory is implemented in available computer code.
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