$$f(z)=\frac{e^{-iz}}{z}$$
What is the value of:
$$\frac{1}{\pi i }\int_{C} f(z) dz$$
if ${C}$ is the arc of the semicircle with radius $R \to \infty$ ,going counterclockwise from point $(R,0)$ to $(-R,0)$.
At first I used the residue theorem and got the correct answer according to the solutions, which is $2$. However i quickly noticed that the curve is a semicircle (angle $0 \to \pi$ instead of $0 \to 2\pi$) so I got confused. I've tried some other things in hope of discovering the way the exercise is supposed to be solved but I honestly just get the feeling it might be something very simple I'm overlooking. As far as I know, Residue Theorem can only be applied when the curve is simple and closed, also the fact that the singularity is at $z = 0$ doesn't help very much.