I am not sure that I am understanding how to find branch points and cuts correctly. So for $\sqrt{z-1}$, I let $z=re^{i\theta+2nx}$ and substitute into $\sqrt{re^{i\theta+2nx}+e^{2kx}}$. But I don't know how to proceed from there.
And how should we find branch points and cuts for logarithm? Say take $log(z^2+z+1)$ for example, should we first substitute it into something like what we've done above? Thanks!
The branch points for $\sqrt z$ are at $z=0$ and $z=\infty $. Therefore, the branch points for $\sqrt{z-1}$ are at $z=1$ (if you like, let $z=1+re^{i\theta}$) and $z=\infty$. The plane can be cut by any contour that adjoins $1$ and the point at infinity. The Principal branch would be the ray along the negative real axis that starts at $z=-1$ and approaches $-\infty$.
The branch points for $\log(z)$ are also at $z=0$ and $z=\infty$. Therefore, the branch points for $\log(z^2+z+1)$ are at $z=\frac{-1\pm i\sqrt{3}}{2}$ and $z=\infty$. Branch cuts may be any contours that begin at the branch points $\frac{-1\pm i \sqrt 3}{2}$ and terminate at $\infty$. Alternatively, a suitable branch cut is any contour that adjoins $\frac{-1+i\sqrt3}{2}$ and ${-1-i\sqrt3}{2}$.