It's commonly used in imaginary-time path integral that "analytic continuation" means replacing $t \rightarrow - i \tau$ or reparametrizing the theory in terms of imaginary time $\tau = i t$. For example, it's common practice to just replace the imaginary time $\tau \rightarrow i t$ to figure out the real time Green's function from the imaginary time Green's function.
I've been having trouble connecting this with the definition from math. My understanding here is to find the analytic continuation of an analytic function in domain $V_1$ around $z_1$: $f(z) = \sum_k a_k (z- z_1)^k$, we need to find $f_2$ defined in another domain $V_2$, which overlaps with $V_1$. And if $f_1 = f_2$ in $V_1 \bigcap V_2 $, then $f_2$ is an analytic continuation of $f_1$ on $V_2$.
How is the replacement justified from the definition in math?
P.S. There is a very similar question here but I don't think the answer was satisfying to me.
P.P.S QMechanic linked a post about Wick rotation, which is showing an integral along the real line is the same as the integral along imaginary axis. One can apply Cauchy theorem under the right conditions, but I thought "analytic continuation" by replacing arguments of $t$ is a strong statement: it's saying 1) there are no poles in the quadrants; 2) the contour integral connecting $t$ and $it$ are zero for all $t$. I suppose I could have missed some simple property about the Green's function, that might imply these two conditions?
