The spectral density can be found very straightforwardly from G R , using the Sokhatsky–Weierstrass theorem lim η η → → 0 + 1 x ± ± i η η = P 1 x ∓ ∓ i π π δ δ ( x ) , where P denotes the Cauchy principal part. This gives ρ ρ ( k , ω ω ) = 2 Im ⁡ ⁡ G R ( k , ω ω ) .