Consider the imaginary time Greens function of a fermion field $\Psi(x,τ)$ at zero temperature
$$ G^τ = -\langle \theta(τ)\Psi(x,τ)\Psi^\dagger(0,0) - \theta(-τ)\Psi^\dagger(0,0)\Psi(x,τ) \rangle $$
It is well known that we can obtain the retarded Greens function by performing Fourier transformation into frequency space and performing the analytic continuation $iω \to ω + i\eta$.
What I would like to do is to perform the analytic continuation directly in the form $iτ \to t$, but I don't know how to deal with the $\theta(τ)$ terms.
How to perform the analytic continuation $iτ \to t$ of the step function $θ(τ)$?
In my case, I am dealing with a chiral Luttinger liquid, giving something like
$$ G^τ(x,τ) = -\left[\theta(τ)\frac i{iλ + ivτ - x} - \theta(-τ)\frac i{iλ - ivτ - x}\right] $$
where $λ \approx 0$ is an infinitesimal but important regularization. Of course, the analytic continuation into the time domain is going to look something like
$$ \frac1{iλ + vt - x} $$
but I'm interested in the precise form.
Also, I'm ultimately interested in the spectral function, so I don't mind if analytic continuation gives me yet another variant of a Greens function, but I would like to obtain it precisely from the imaginary time Greens function without going through a tedious Fourier transform. For instance, Giuliani and Vignale's book "Quantum Theory of the Electron Liquid" uses the Greens function $G_{>}(x,t)$ to great effect (equation (9.133)).
