As preparation for my exam I "invented" the following problem as an exercise to see whether I understand how to work with branch points.
$f(z) = \frac{z}{\sqrt{z^2+1} (z^2 +a^2)}$
The goal is to calculate $\int_0^{\infty} f(z) dz$.
I made the following conventions (see figure) for my branches.
$\theta_1 \in [-\pi/2, 3\pi/2)$ $\theta_2 \in [\pi/2, 5\pi/2)$ (let $I_i$ be be the integral on contour $C_i$) I'm able to calculate that (using my conventions)
$I_1=I_5$
$I_2=-I_4$
$I_6 = 0$
But I'm unsure what $I_3$ is equal to.
My question is: How do I calculate $I_3$?
Update 1
My difficulty lies in the fact that $theta_2$ is going to "jump" if I would just integrate in a circle with radius going to zero. Should I maybe do 2 semi-circles? So that I can choose the right value for $theta_2$?
D