Thanks! I'm not familiar with regularization, but it seems if I were to look into a book or internet material on QFT I'd find some explanations of it. I'll also have to read up on the saddle point method. Just to make things explicit, my understanding of how the saddle-point method can accommodate imaginary stationary solns is by turning the integral over $R$ into a contour by connecting the real line and the steepest descent path through the stationary soln. Assuming there are no poles inside the contour, then the real integral will be equal to the integral over the steepest descent path?

@Nanite I acknowledge this difficulty. I'm mostly just going through this book and trying to understand the leaps the author makes. He uses this technique (Fourier Transform of the delta function -> saddle point approximation) many times in the book, so it seems like it might be an important technique to understand. Furthermore, demonstrating ensemble equivalence can be a useful exercise in developing facility with the tools of stat mech.

This might help: physics.stackexchange.com/q/18576