In the calculation of Mercury using perturbation, we have to evaluate the integral: $$ \delta \phi =-\frac{GMm}{2a^3}\sqrt{\frac{m}{2E}}\frac{\partial}{\partial J}\int^{r_{max}}_{r_{min}}\frac{r^3 dr}{\sqrt{(r-r_{min})(r_{max}-r)}} ;$$ then I found that in the textbook, it uses the contour integral: $$ I =\int^{r_\max}_{r_\min}\frac{r^3 dr}{\sqrt{(r-r_{min})(r_{max}-r)}} =\frac{1}{2}\sqrt{u_1u_2}\oint\frac{du}{u^4\sqrt{(u-u_2)(u_1-u)}}; $$ where $u_1=1/r_{min}, u_2=1/r_{max}$.
I just wonder why it can be changed to contour integral. We know $u_1$ and $u_2$ are the singularities and the branch point, but why the integral along the real axis of $u$ are the two line one up the real axis and one below, I cannot figure out.
Thanks for answering my question!