This furthermore implies that G R ( k , ω ω ) obeys the following relationship between its real and imaginary parts: Re ⁡ ⁡ G R ( k , ω ω ) = − − 2 P ∫ ∫ − − ∞ ∞ ∞ ∞ d ω ω ′ 2 π π Im ⁡ ⁡ G R ( k , ω ω ′ ) ω ω − − ω ω ′ , where P denotes the principal value of the integral.