Integration question in Peskin and Schroeder

[ianhoolihan]

Hi all, I'm stuck with proving the last step of (2.51) in Peskin and Schroeder:
$$\begin{align} D(x-y) &= \frac{1}{4\pi^2}\int^\infty_m dE \sqrt{E^2 - m^2}e^{-iEt}\\
& \approx_{t \to \infty}\ \ e^{-imt}\end{align}$$

I've read on another post that the solution is to use the method of stationary phase, but I do not see how this applies, as $E$ is not a rapidly oscillating function...?

Thoughts appreciated,

Ianhoolihan

 

[strangerep]

You didn't say which "other thread" you read, but my post #13 in this thread might shed a little more light...

https://www.physicsforums.com/showthread.php?t=233950

 

[ianhoolihan]

Thanks strangerep,

The thread I referred to was https://www.physicsforums.com/showthread.php?t=424778.

Your post does make it clearer, in that it stems from a limit of the exact solution (Bessel function). I will look through the details soon.

Cheers.