I have the following integral:
$$ \int_{-\infty}^{\infty} \frac{\mathrm d\tau}{2\pi \mathrm i} \int_{-\infty}^{\infty} \frac{\mathrm d\tau'}{2\pi \mathrm i} \frac{\mathrm e^{-c^2(\tau - \tau')^2}}{(\tau - \mathrm i \epsilon)(\tau' - \mathrm i\epsilon')}. $$
Here I wanted to do the $\tau'$ integral first, so I wrote the following
$Assumptions = c >= 0 && c \[Element] Reals && ϵ > 0 && ϵ \[Element] Reals && δ > 0 && δ \[Element] Reals && τ \[Element] Reals && t \[Element] Reals;
I5 = Integrate[E^(-c^2 (τ - t)^2)/(( τ - I ϵ) (t - I δ)), {t, -Infinity, Infinity}]
where I have denoted $\tau'$ using t. However the output turns out to be just the same expression as the input.
Next I tried to use the residue of the integrand about $ \tau' = \mathrm i\epsilon' $ (In the code below, about $ t = \mathrm i\delta $).
Residue[e^(-c^2 (τ - t)^2)/((τ - I ϵ) (t - I δ)), {t, I δ}]
Again the output turns out to be the same expression as the input.
Any suggestions how to compute the integral?
EDIT: It was pointed out in the comments that there were some mistakes in the code, so I edited the same.
$Assumptions = c>=0 && c \[Element] Reals && \[Epsilon] > 0 && ...
$\endgroup$Integrate[E^(-c^2 (τ - t)^2)/((τ - I ϵ) t), {t, -Infinity, Infinity}, Assumptions -> {c > 0 && ϵ > 0 && τ \[Element] Reals}, PrincipalValue -> True]
. The imaginary part is trivial to evaluate using $\mathrm{im}\left[\frac{1}{x-i\epsilon}\right]=\pi\delta(x)$. $\endgroup$