I have a question about contour integral containing two branch points.
$$\oint\frac{z^2}{(-iz^5-a)^{5/2}}$$
There are 5 branch points in total and I want to choose the positive imaginary axis as the branch cut (red line), and the contour is plotted in purple dash line. This contour contains two branch points A and B. Take the A's part as an example: L1 and L2 are very close to the OA($z=re^{i\pi/10}$) segment. $L1=re^{i\pi/10}-iw$ and $L2=re^{i\pi/10}+iw$. I think the integral on Ra will be 0 once w approaches 0. Is it real that the contour integral equal to $2\int_O^A\frac{z^2}{(-iz^5-a)^{5/2}} dz$? Figure is here
Any help in understanding this integration will be greatly appreciated!