I am doing a physics problem and have come across a contour integral that I just don't know how to solve. I do not have the complex analysis background and I am wondering if anyone can explain how to do it from a beginners perspective. I have done a lot of research on my own but I am still confused about a few things.
The integral is the following.
\begin{equation} J=\frac{\sqrt{2m}}{2\pi}\oint \sqrt{X_0-a\sec^2\left( \frac{x}{l} \right)}dx \end{equation}
My professor says I need to look at "the branch points as well as at infinity." So I was able to determine that if I write this as a series and take the coefficient of the term with exponent -1, that will be the residue, (Not sure why, again I don't have much expeirence with this stuff.) and then I can use the residue theorem. When I did this, I got a residue of $i\sqrt{a}$, meaning the integral should be $\frac{\sqrt{2m}}{2\pi}2\pi i (i\sqrt{a})=-\sqrt{2ma}$. But this can't be the whole picture as I didn't do anything with branch cuts. (I also know this because I absolutely need an $X_0$ term by virtue of the physics problem I am doing) I am not even sure what they are honestly. After reading a few things I determined that they seem to be the location where the function gets defined in a piecewise fashion. But I am confused as to how to use them for evaluating an integral. I have taken a look at other examples on the internet but I always seem to get confused while going through the process they show. What will help me the MOST is to solve THIS integral, since it is the one I spent so much time on.
If anyone can help me in any way it would be much appreciated!
EDIT: I believe I have determined the contour of the problem. I needed to take into account the positive and negative nature of the square root and I get the following contour (just the center loop)
