Show that $\int_{C_R} \frac{z e^{iz}}{1+z^2}dz$ where $C_R$ is the half circle in the upper half plane with radius $R$ tends to $0$ as $R$ goes to infinity.
My professor showed me something with Jordans Lemma here but I can't really remember what he did. Why can't I do it just like this:
$$ \left| \int_{C_R} \frac{z e^{iz}}{1+z^2}dz\right|≤ \int_{C_R} \frac{\left| z e^{iz} \right|}{\left|z^2\right|-1} \left|dz\right|≤ $$
{Now for $\left| e^{iz} \right|≤1$ in the upper half plane so we have that}
$$ ≤ \frac{R}{R^2-1} $$
and this goes clearly to zero as $R$ goes to infinity.
What have I missed?