Let $g(z)$ be a branch of the square root on $\mathbb{C} \setminus \lbrace iy : y \leq 0 \rbrace$.
For $0 < r < 1 <R$ and $0 \leq \theta \leq \pi$, let $\tau_r$ be the contour given by the semi-circles $re^{i \theta}$, $Re^{i \theta}$ and the line segments on the real axis $[-R,-r]$, $[r,R]$.
How can we evaluate the following?
$$\int_{\tau_r} \frac{g(z)}{z^2 + 1}$$
I'm asking this as the example we worked out in class had a trigonometric function for $g(z)$. I don't get how that example can guide me in this one. I'd appreciate step-by-step explanation if possible, as I'm really lost on this.
Don't you just calculate residues at $i,-i$?
When $|z|=R>1$, the integrand is less than $\frac{\sqrt{R}}{R^2+1}$, thus the modulus of the whole integral is less than $\frac{\pi R \sqrt{R}}{R^2+1}$. We have $\lim_{R \rightarrow \infty} \frac{\sqrt{z}}{1+z^2} = 0$ (also holds for $r$)