# Convergence Property of Path-Integral

Let the action be $$S= \int \bigg\{ \frac{1}{2} \big(\frac{dX}{dt}\big)^2 - V(X) \bigg\} d\tau$$

and the corresponding Path-Integral

$$Z= \int DX(t) e^{iS}.$$

Since the convergence is not clear we Euclideanize the time coordinate $$t$$ by the Wick rotation

$$t \rightarrow -i \tau$$

and get the Path-Integral

$$Z_E=\int DX(\tau) e^{-S_E},$$

with $$S_E= \int \bigg\{ \frac{1}{2} \big(\frac{dX}{d\tau}\big)^2 + V(X) \bigg\} d\tau.$$

And now my question - the Euclideanized path-Integral allegedly has a better convergence property, but i do not quite see why this is the case?

In a nutshell, the Euclidean Boltzmann factor is exponentially suppressed because the Euclidean action is bounded from below (assuming the potential $$V$$ is bounded from below), while the Minkowskian Boltzmann factor has modulus 1 and is oscillatory.