I have to classify the singularity at the origin of the function $f(z)=\frac{1}{z} + \frac{1}{\sin z}$. How do I get this answer without passing through the Laurent serie directly . Is anyone could help me at this point?
2026-03-28 04:28:16.1774672096
Classification of singularities - $f(z)=\frac{1}{z} + \frac{1}{\sin z}$
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Multiplying by $z$: $$ z f(z) = 1 + \frac{1}{\operatorname{sinc} z}, z \neq 0 $$ $\operatorname{sinc} 0 = 1$ and hence the above function has a removable singularity at $0$, so $f(z)$ has a simple pole at $0$.