Imaginary Time and Path Integrals?

[LarryS]

Steven Hawking writes in A Brief History of Time that time itself must sometimes have an imaginary component in order for Feynman's Sum-Over-Histories approach to work. Why, in a nutshell, is this so? Thanks in advance.

 

[Avodyne]

Convergence of the integral.

 

[PRB147]

It is wick rotation, please google "wick rotation",
you also can find wick rotation in Wiki