  We derive simple formulas connecting the generalized Wigner functions for
$s$-ordering with the density matrix, and vice-versa. These formulas proved
very useful for quantum mechanical applications, as, for example, for
connecting master equations with Fokker-Planck equations, or for evaluating the
quantum state from Monte Carlo simulations of Fokker-Planck equations, and
finally for studying positivity of the generalized Wigner functions in the
complex plane.
