Classifying the singularities of a function

Question

I'm trying to classify the singular points of the following function but I'm having some troubles.

$$f(z) = \frac{z(z-1)^{1/3}\sin(1/z)}{(z^2+1)^3\sin(z)}$$

This is what I have so far:

1. $z=i, z=-i$ are poles of order 3 because of the third power in the denominator.
2. Because of the $\sin(z)$ we have simples poles for $z=n\pi, n \in Z$
3. I suspect that $0$ is also a singularity because of the $\sin(1/z)$ but I'm not sure.

What do you think ?

Answer 1

The first two point are correct.

1. $z=i, z=-i$ are poles of order 3 because of the third power in the denominator.
2. Because of the $\sin(z)$ we have simples poles for $z=n\pi, n \in Z$

Now recalling $sin(\frac{1}{z})= Imm(e^{\frac{1}{z}})$ by developing in series $sin(1/z)=\sum_{k, dispari}(i)^{k-1}(\frac{1}{z})^k(\frac{1}{k!})$ Intuitevely $sin(1/z)=\frac{1}{z}-\frac{1}{6z^3}+...$. Which is by definition an essenzial singularity.