I find some difficulty in understanding the limits of the two integral below (on Page 27 of Peskin & Schroeder's Quantum Field Theory): $$D(x-y)=\frac{1}{4\pi^2}\int_m^\infty d E \sqrt{E^2-m^2}e^{-i Et}\sim e^{-imt}|_{t\rightarrow \infty},$$ where 4-vector $(x-y)$ only have time-component, space ones vanish (I omitted one step on the above deduction, because I don't think it's the main purpose here); $$D(x-y)=\frac{-i}{2(2\pi)^2r} \int_{-\infty}^\infty dp\frac{p e^{ipr}}{\sqrt{p^2+m^2}}\sim e^{-mr}|_{r\rightarrow \infty},$$ where in the later correlation function, we set $\vec{x}-\vec{y}=\vec{r}$, and time-component of $x-y$ vashnies.

My only question is how to get the intended approximation above?

If you have some time, please help me out, because I have run into these kind of asymptotic approximations many times and don't know how to deal with it.

  • $\begingroup$ I know these integrals deal with physics, but perhaps this question would be better on Mathematics SE? The heart of your question just deals with evaluating integrals when a parameter goes to infinity. Your question is not about physics concepts. $\endgroup$ Feb 6 '19 at 23:22
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    $\begingroup$ See physics.stackexchange.com/questions/307856/… for your second integral. $\endgroup$
    – wcc
    Feb 6 '19 at 23:29
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    $\begingroup$ Actually the above link answers your first integral as well. $\endgroup$
    – wcc
    Feb 6 '19 at 23:49