  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.
