How to solve this Complex Integral using poles?

Question

I want to find the green's function of a free particle, which depends of the integral: $$ I = \frac{1}{4\pi ²ir} \int^{+\infty}_{-\infty} \frac{ke^{ikr}}{E-\frac{\hbar²k²}{2m}+i\eta} dk\,. $$

Then, using the cauchy principal value we remove the $\eta$...The result is as follows:

$$ g=\frac{e^{i\frac{\sqrt{2mE}}{\hbar}r}}{4\pi r} $$

Answer 1

I'll give you some hints:

1. Get your integral in more combined form: $$ I(r) = \int \frac{k e^{i k r}}{A - k^2} $$ here $A$ is constant. Forget about everything else till the end.
2. Think of the function $F(r)$, s. t. $$ F'(r) = I(r) $$
3. Draw some contours and see which one is better for $F(r)$. If nothing comes to mind, try literature, I would recommend: Russell L. Herman, An introduction to mathematical physics via oscillations
4. Now return to your preintegral constant and substitute A.