I'm trying to prove that the following contour integral approaches 0 as R -> $\infty$. How exactly would we go about doing this? $$ \int{\log\left(z^{2} + 1\right) \over 1 + z^{2}}\,{\rm d}z\quad \mbox{over the curve}\quad z=Re^{i\theta}\quad \mbox{from}\quad \theta=0\ \mbox{to}\ \theta={\pi \over 2}. $$ Usually, I use inequality arguments to prove this, but they're not working very well ( the magnitude of the logarithm messes it up ).
How are we showing that this integral approaches $0$ as $R \to \infty$ ?.
