Simple Complex Integral from Peskin and Schroeder Eq. (6.4)

Question

In the 6th Ch. of the QFT book by Peskin and Schroeder, there is a very simple integral which gives the result as in eq. (6.4). The integral is:

$$\int_{0}^{\infty} d\tau(\frac{p'^{\mu}}{m}) \exp^{i(kp'/m+i\epsilon)\tau}=i(\frac{p'^{\mu}}{kp'+i\epsilon}).\tag{6.4}$$

I tried to solve this integral, I do not understand how the $\tau=\infty$ part converges.

Answer 1

Note that $$ \int_0^\infty d\tau e^{i(kp'+i \epsilon)\tau} = \frac{1}{i(kp'+i \epsilon)} \left( \lim_{\tau \to \infty} e^{i(kp'+i \epsilon)\tau}-1\right). $$ But we have $\lvert e^{i(kp'+i \epsilon)\tau}\rvert = e^{-\epsilon \tau}$, so it converges to zero as $\tau \to \infty$. This $\epsilon$ is added precisely for this reason.