Let $a$ be a point on the complex plane such that the function $f$ has an essential singularity. I am trying to prove that the square of that function also has an essential singularity.
I suppose that it does not have an essential singularity. Then $\exists ;k$ such that $(z-z_0)^{2k}f^2(z)$ differentiable. Then if I could take the square root and preserve the differentiability I would have finised. Can I do that?
Suggestion. Another, more convenient way to test if an isolated singularity of $f(z)$ at $z_0$ is not essential: $ (z- z_0)^k f(z) \to L$ for some $L$ in the extended complex plane.