 This implies that the density matrix should be in "anti"-normal order so that we can express the density matrix as a power series

Inserting the identity operator
we see that
and thus we formally assign

More useful integral formulas for are necessary for any practical calculation.One method is to define the characteristic function
and then take the Fourier transform

Another useful integral formula for is

Note that both of these integral formulas do "not" converge in any usual sense for "typical" systems .We may also use the matrix elements of formula_2 in the Fock basis formula_13.