This question comes from physics...

We have the following functional (or function dose not matter): $$S\left[x\right]:=\intop_{-\frac{t_{0}}{2}}^{+\frac{t_{0}}{2}}dt\,\mathcal{L}\left[x\right]\::\:{\displaystyle \mathcal{L}}:={\displaystyle \frac{m}{2}\left(\frac{dx}{dt}\right)^{2}-V\left[x\right]}$$

Religiously speaking, how to do an analytic continuation for $S$?

Usually, we follow this by the so called Wick rotation $t\rightarrow-i\tau$, this leads to get $iS=-S_{E}$ where: $$S_{E}:={\displaystyle \intop_{-\tau_{0}/2}^{+\tau_{0}/2}\mathcal{L}_{E}\,d\tau}\::\:\mathcal{L}_{E}\left[x\right]:={\displaystyle \frac{m}{2}\left(\frac{dx}{d\tau}\right)^{2}+V\left[x\right]}$$

While it is clear how to do the Wick rotation, It seems to me that I am missing something in relation to integral boundaries, where somehow $i$ disappears there after Wick rotation, I think the reason burred in how to analytically continue $S$ at first place.