I wish to calculate the integral $$\int_{|z|\mathop=2}\sqrt{z^4-z}\,dz$$
$$z^4-z=z(z-1)(z-e^{2i\pi/3})(z-e^{-2i\pi/3})$$ We must make branch cuts which go through the branch points. The branch points of $\sqrt{z^4-z}$ are $z=0,1,e^{2i\pi/3},e^{-2i\pi/3}$. All of these lie inside this contour, so I first thought that I must work out the residue at each of these points, and then use the residue theorem. However there is no nice way that I could think of to calculate these residues.
Then it occurred to me that I am not even sure if they exist - these are not poles, they are branch points with branch cuts through them. I then saw that the question has a hint saying to use Laurent Series. Most of the Laurent series I have ever seen have been functions with (sort of) Taylor series, but with some change of variables, or multiplied by some factor of $1/z^n$. The point is - I don't know any (efficient) way of finding the Laurent Series for this function.
My questions: How can I
Define these branch cuts,
Find the different Laurent Series, and find the value of the coefficient of $1/z$, thus finding value of the integral?
Also:
I thought Laurent Series existed within an annulus of the point about which they exist - how is this possible for a function with branch cuts? Is there something I am confusing here? These two ideas I had seem to contradict.
If the Laurent Series do exist, then is it possible to calculate the residues directly? (Also assuming these exist)