Does the function $f(z) = \sin(\sqrt{z})$ have a branch point at infinity? I'm confused because infinity is an essential singularity of $\sin(z)$, so I'm not sure how to do the usual $z\to w=1/z$ substitution and analyse the resulting function $f(w)$ at zero... Any help would be appreciated.
2026-03-25 20:41:56.1774471316
Is infinity a branch point of $\sin(\sqrt{z})$?
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$\frac{\sin(z^{1/2})}{z^{1/2}}$ is entire,
$\frac{\sin(z^{-1/2})}{z^{-1/2}}$ has an isolated (essential) singularity at $0$
thus, $\sin(z^{-1/2})$ has a branch point at $0$ and $\sin(z^{1/2})$ has a branch point at $\infty$.