Doubt Let $f(z)=\mathbb{tan 1/z }$ then what kind of singularity of $f(z)$ at $z=0?$
My approach is $f(z)=\mathbb{tan1/z}=\frac{\mathbb{sin1/z}}{\mathbb cos1/z}$. Now we can easily check that the set zeros (say $S_1$) of $f(z),$ $S_1=\{1/n\pi \mid n \in \mathbb{Z-\{0\}}\}$ and the set of poles (say$ ~S_2)$ are $S_2=\{2/(2n+1)\pi \mid n \in \mathbb{Z}\}$ and clearly $0$ is limit point of $S_1$ and $S_2$.
and we know the limit point of zeros of an analytic function is an essential singularity unless the function is identical to zero and the limit point of poles of an analytic function is a non-isolated essential singularity.
Note: Can we say a point is a non-isolated essential singularity if it is a limit point of singularity for example $f(z)=\mathbb{tan1/z}$, $\mathbb{log z}$ on the negative real line (because non-isolated singularity depends on other singular points as well) and if it is not a limit point of other singularities then it is isolated essential singularity, for example, $f(z)=e^{1/z}$.
Please correct me if I am wrong.
Thanks in advance!