# How to grasp the limits of these two integrals? [duplicate]

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.

• 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. – BioPhysicist Feb 6 '19 at 23:22
• See physics.stackexchange.com/questions/307856/… for your second integral. – wcc Feb 6 '19 at 23:29
• Actually the above link answers your first integral as well. – wcc Feb 6 '19 at 23:49