Let $f(z)=(e^{3iz}-3e^{iz}+2)/z^3$, this clearly has a singularity at $z=0$, how do you show that this is a simple pole? i.e. a pole of order one, every way it definity expands to something with a minimum power of $-3$, so I come to the conclusion there is a pole of order $3$?
And if this is the case is $f(z)+3/z$ holomorphic? Does this term cancel some part of the expansion I have missed that eliminates the pole?