Recently I've been working on branch cuts of square root functions, and come to problems like this:
Find a single-valued analytic branch $f$ of $\sqrt{z^2+z}$ on the set $\{ z \in \mathbb{C}: |z| >1 \}$ such that $f(2) = -\sqrt{6}$.
Evaluate the integral of $f$ around the contour $|z| =2$ (positively oriented) using the Laurent series of the square root.
I can find an $f$: the function $-\sqrt{z}\sqrt{z+1}$ where the square roots are the principal branch of the square roots function.
However, the only example of using Laurent series I've seen (in these notes) for contour integrals like this works by factoring a $z$ out of a square root; i.e. writing
$$ (z^2+z)^{1/2} = z(1+1/z)^{1/2} $$
and then using the Laurent expansion for the (principal branch) square root $\sqrt{z}$ about $z=1$ to expand $(1+1/z)^{1/2} $.
My question is about the signs here. The approach above assumes that $\sqrt{z^2} = z$, but it seems that in my branch of the square root, I have $\sqrt{z^2} = -z$.
And then the square root I'm left with in the factorization is still defined based on the branch of $\sqrt{z^2+z} \;$ I'm using, so would I use the Laurent expansion for $- \sqrt{z}$ instead?
I understand that if both signs need to be negative, there's no net effect on the result; I'm just trying to understand the details of the computation. Any insight would be appreciated.
Best Answer
Concretely expand $\sqrt{z^2+z}=-z\sqrt{1+1/z}\ $ that verifies $f(2)=-\sqrt{6}$ (the $+$ possibility doesn't apply).
The cut should be chosen between the points $-1$ and $0$ as in your linked file because $f(2)=-\sqrt{6}$ is indicated without additional specification (compare to 'the top side of the cut' in the pdf) so that the function must be smooth at point $z=2$. This point would indeed be on the cut had we chosen the cut outside of $(-1,0)$.
Here is a picture of the argument of your function as considered (the minus adds $\pi$ to the argument) :
note that for $z$ real with $z<-1$ the arguments become equal to 0 $\pmod {2\pi}$
The same kind of problem was handled in this other thread with the cut shown outside the points or inside. Note that there is a link to the handling of the general case at the end (just set a=−1 and b=0 there).