Yea I don't understand Trimok's comment:
"@DanielSank : the iϵ Feynman prescription, in your case p2−m2→p2−m2+iϵ, is done to be sure of the convergence of the QFT partition function Z=∫DΦeiS(ϕ), with S(ϕ)=−ϕ(□+m2)ϕ, so this has the effect to add a term e−ϵϕ2 in the integral, making the integral convergent. An other point of view is that, with the Feynman contour, you may do a Wick rotation (90° rotation), going to a imaginary (euclidean) energy , without crossing the poles. – Trimok"