I'm currently following a derivation of the retarded Greens function in my Electrodynamics course.
We arrived at a integral of the following form: $$\int_{-\infty}^{\infty}\mathrm d\omega\frac{1}{c^2k^2-\omega^2}e^{-i\omega\tau}$$
with poles at $\omega=\pm ck$.
The poles are located on the real-axis and we choose an integration path such that the two poles are by-passed in the upper half-plane.
Now my book says that if we have $\tau>0$ we have to close our integration path on the lower half-plane and if $\tau<0$ we have to close it in the upper half-plane so that the exponential doesn't blow up at infinity.
I don't really get why. Because if we have $\tau>0$, the exponent stays negtive so the imaginary part of $\omega$ would need to go to $+\infty$ for $e^{-i\omega\tau}$ to not blow up. So we would need to close our half circle in the upper half-plane. The same reasoning applies for the case $\tau<0$.
It would be great if someone could point out my mistake here. Appreciate any advice, thanks!