Four dimensional contour integration

Question

I know that the topic is related with physics, but the very difficulties in it arise from a purely mathematical computation. Nobody could help me in the PSE section, hope to find any help from mathematicians.
In the book N. D. Birrell and P. C. W. Davies, "Quantum Fields in Curved Space" at pp. 52-53 the four dimensional (positive frequency) Wightman Green function is expressed in position space as $$ D^{+}(x,x') = -\frac{1}{4\pi[(t-t'-i\epsilon)^2 -|\vec{x}-\vec{x}'|^2}.\tag{3.59}$$ It is said that this result comes from solving the momentum space integral $$\frac{1}{(2\pi)^4}\int d^4p\frac{1}{p_0^2 - |\vec{p}|^2}e^{-ip_0(t-t')+i\vec{p} ({\vec{x}-\vec{x}')}}$$ with the specific contour reported in figure 3 at p. 22 (a circle arount the positive pole). Anyway, I cannot find the above result. Instead, I find something very similar to the retarded Green function, that depends only from $\frac{1}{4\pi|\vec{x}-\vec{x}'|}$ and a $\delta$-function which fixes the time. I think that I'm getting wrong on the evaluation of something involving the $i\epsilon$ prescription used to shift the poles. Can anybody help me getting through this? I can also add my trials if needed.