Find all singular points of f(z), classify them and inspect the behaviour at infinity

Question

Find all singular points of f(z), classify them and inspect the behaviour at infinity

$f(z)=sin(1/z)+1/z^2$

Here i'm stuck to find singularities since z=0'd be a singularity however we have $+sin(1/z)$ so i guess z has to be a value to make whole function singular. Then i realized i may be rewrite this function as a series expansion but i failed. So i need some suggestions or guidance.

Answer 1

You've got the right idea.

Note that since you need to inspect the behavior at infinity, you'll want to consider the function $$g(z):=f\left(\frac1z\right)=\sin(z)+z^2.$$

I would start by finding the series expansion for $g$ about $0,$ then noting that $f(z)=g(1/z).$ That will help you show that $z=0$ is the only singularity of $f,$ as well as to classify it.