Contour integral example from "QFT for the gifted amatueur"

[marcom]

Hi,

I've never studied compex analysis before but I am trying to understand this example from "QFT for the gifted amatur".
I don't understand why the residue at the pole is e^-iEp(t-t')e^-e(t-t'). How did the find e^-e(t-t')?

Thanks.

 

[stevendaryl]

Well, he (or she) starts with the function:

$\dfrac{i e^{-i E(t - t')}}{E - E_p + i \epsilon}$

The residue of a function of the form $\dfrac{A(E)}{E-K}$ is just $A(K)$. In this particular case, $K = E_p - i \epsilon$, and $A(E) = i e^{-i E(t-t')}$, so the residue is $i e^{-i (E_p - i \epsilon) (t - t')} = i e^{-i E_p (t-t')} e^{- \epsilon (t-t')}$

 

[marcom]

Thanks!