Does the multi-valued function $\sinh^i(z)$ have an analytic branch on the upper half plane?

Question

I've tried algebraically manipulating the function but I'm stuck as to how to continue. $$\sinh^i(z) = \left(\frac{e^z -e^{-z}}{2} \right)^i = e^{i \log \left(\frac{e^z -e^{-z}}{2} \right)}$$ I then tried assuming I'm looking at the principal branch of log and work backwards to see which parts of the image correspond to those of the source but I only really did it with Mobius transforms and I'm not good at those.

Answer 1

You are on the right track, but you may do better by looking at the branch points of $f(w)=w^i$ where $w=\sinh z$.

Since this $f(w)$ has a branch point whenever $w=\sinh z=0$, we conclude that these exist at all points given by $z=i(k\pi)$; $\sinh z = -i\sin(iz)$ is zero at such points. Therefore an analytical branch in the upper half-plane (or the lower one) is not possible.

However, the alignment of all branch points along the imaginary axis does imply that an analytic branch may be defined in the right or left half-plane.