I have an integral of the type:
\begin{equation} \mathcal{I}=\kappa \int_{-\infty}^t d\tau e^{-\kappa |t-\tau|} e^{f(t-\tau)} \end{equation} where $f(t-\tau)$ is a generic function such that $f(0)=0$. I am interested in two limits:
$\bullet$ $\kappa \rightarrow \infty$
In this case, is it correct to assume $e^{-\kappa |t-\tau|}\rightarrow \delta(t-\tau)$? If it were, such integral becomes \begin{equation} \mathcal{I}=\kappa/2 \end{equation} where we have used that $\int_{-\infty}^t\delta(t-\tau) d\tau=1/2$ because of the upper limit.
$\bullet$ $\kappa \rightarrow 0$
In this case, we cannot assume $e^{-\kappa |t-\tau|}\rightarrow \delta(t-\tau)$ anymore and we have to deal with the full integral. While there can be cases where it would be a problem, it is for me. Specifically, my function $f(x)$ is a superoperator, or in other words, a matrix. Are there any approximations I could make here? Some Taylor expansion around $\kappa=0$? Are they justified?