Finding all the possible values of an Integral in the Complex Plane I am studying Complex Analysis by Lars V Ahlfors. I am unable to solve one of his exercises. It is:
Find all possible values of $$\int \frac{dz}{\sqrt{1-z^2}}$$
over a closed curve.
I do not have any clue as to how to solve the above problem, if the denominator terms like $z-z_0$ then I can try winding a curve around $z_0$. But in this in particular question I cannot think of any way to approach it. If anyone can give some hints it would be great. Thanks.
 A: The exercise considers regions in the complex plane such that the two points $1$ and $-1$ belong to the same component of the complement. On such regions, an analytic branch of $\frac{1}{\sqrt{1-z^2}}$ exists, which is the first part of the exercise, and the second part asks for the possible values of
$$\int_\gamma \frac{dz}{\sqrt{1-z^2}}$$
where $\gamma$ is a closed curve (suitable for integration) in such a region.
The two most used regions to define branches of $\frac{1}{\sqrt{1-z^2}}$ are:


*

*$U = \mathbb{C} \setminus \{ t\in\mathbb{R} : \lvert t\rvert \geqslant 1\}$, where $1$ and $-1$ belong to the same component of the complement of $U$ in the sphere $\hat{\mathbb{C}}$, but not to the same component of the complement in the plane, so it is arguable whether this region or similar ones are to be considered. $U$ is simply connected, so Cauchys integral theorem immediately yields the vanishing of all the integrals in that case.

*$V = \mathbb{C} \setminus [-1,1]$, and in this case it is easy to see that the value of the integral only depends on how often $\gamma$ winds around the interval $[-1,1]$, so the value of the integral is $n(\gamma,1)\cdot C$ for some constant $C$, which I leave for you to find.
Now what remains is to see that every case can be reduced to one of the above. Since there are no assumptions on the niceness of the complement, the argument, when made precise, is a little tedious (Hint: the trace of the curve $\gamma$ is a compact set, hence has a positive distance from the complement, which allows you to replace the complement with a nicer set).
