[tex]S[x(t)]=\int_0^\infty dt'\left[\frac{m}{2}\left(\frac{dx}{dt'}\right)^2-V(x)\right][/tex]

After the so-called wick's rotation[tex]t'=-i\tau[/tex] with [tex]\tau[/tex] being real,

the action becomes

[tex]S[x(t)]=i\int_0^\infty d\tau\left[\frac{m}{2}\left(\frac{dx}{d\tau}\right)^2+V(x)\right][/tex]

my question is why the upper limit of the integral in the secong equation is still [tex]\infty[/tex]?

I think ist should be [tex]+i\infty[/tex]