レーマン表示
G A B ∓ ( ω ) = 1 2 π ∫ − ∞ ∞ [ 1 ω − ω ′ + i η ∓ e − β ħ ω ′ ω − ω ′ − i η ] S A B ( ω ′ ) d ω ′ = 1 2 π ∫ − ∞ ∞ [ 1 ω − ω ′ + i η ∓ e − β ħ ω ′ ω − ω ′ − i η ] ( 1 ∓ e − β ħ ω ′ ) − 1 ρ A B ± ( ω ′ ) d ω ′ = 1 2 π ∫ − ∞ ∞ [ P 1 ω − ω ′ − i π { tanh  ( β ħ ω ′ 2 ) } ∓ 1 δ ( ω − ω ′ ) ] ρ A B ± ( ω ′ ) d ω ′ {\displaystyle {\begin{aligned}G_{AB}^{\mp }(\omega )&={\frac {1}{2\pi }}\int _{-\infty }^{\infty }{\Bigg [}{\frac {1}{\omega -\omega '+i\eta }}\mp {\frac {e^{-\beta \hbar \omega '}}{\omega -\omega '-i\eta }}{\Bigg ]}S_{AB}(\omega ')d\omega '\\&={\frac {1}{2\pi }}\int _{-\infty }^{\infty }{\Bigg [}{\frac {1}{\omega -\omega '+i\eta }}\mp {\frac {e^{-\beta \hbar \omega '}}{\omega -\omega '-i\eta }}{\Bigg ]}(1\mp e^{-\beta \hbar \omega '})^{-1}\rho _{AB}^{\pm }(\omega ')d\omega '\\&={\frac {1}{2\pi }}\int _{-\infty }^{\infty }{\Bigg [}{\rm {P}}{\frac {1}{\omega -\omega '}}-i\pi {\Bigg \{}\tanh {\Big (}{\frac {\beta \hbar \omega '}{2}}{\Big )}{\Bigg \}}^{\mp 1}\delta (\omega -\omega '){\Bigg ]}\rho _{AB}^{\pm }(\omega ')d\omega '\end{aligned}}}
G A B R ∓ ( ω ) = 1 2 π ∫ − ∞ ∞ ρ A B ± ( ω ′ ) ω − ω ′ + i η d ω ′ {\displaystyle G_{AB}^{R\mp }(\omega )={\frac {1}{2\pi }}\int _{-\infty }^{\infty }{\frac {\rho _{AB}^{\pm }(\omega ')}{\omega -\omega '+i\eta }}d\omega '}