I have the complex function $f(z)=(z^2+1)^{1/2}$ which had branch point singularities at z=i, z=-i.
To try to find the value of $f$ on either side of the cut I considered $z=i+\epsilon e^{i\theta}$, which gives
$f=(2i\epsilon e^{i\theta}+\epsilon ^2 e^{2i\theta})^{1/2}$.
I then let $\theta --> \theta +2\pi$. I was hoping I would then see a nice factor come out of the expression for $f$ to see how it changes. But i get
$f=(\epsilon e^{i\theta+\pi}+\epsilon ^2 e^{2i\theta + 4\pi})^{1/2}$
and I don't know what to do with this...
Perhaps I am missing something obvious or made a silly error, or maybe my approach is completely wrong?