The similarity of the spectral representations of the imaginary- and real-time Green functions allows us to define the function G ( k , z ) = ∫ ∫ − − ∞ ∞ ∞ ∞ d x 2 π π ρ ρ ( k , x ) − − z + x , which is related to G and G R by G ( k , ω ω n ) = G ( k , i ω ω n ) and G R ( k , ω ω ) = G ( k , ω ω + i η η ) . A similar expression obviously holds for G A .