i was told here that there is no +infinity and -infinity in complex functions
but i have a solved question from a test which is solved exactly like it you said was wrong
and got full credit
maybe i am missing something here
As an example of what you're doing wrong take . Clearly we can think of it as a real or complex function. Analyze the behaviour in both cases and remember why we can talk about poles.
Frankly I wouldn't know how to turn that expression into a Laurent series, but it's not necessary, that is why one develops propositions like: if with open, holomorphic and has a removable singularity at and has an essential singularity at then has an essential singularity at
Basically, because near a pole, a function behaves like for some integer m. Try to work out the details by yourself (maybe using the multiplication formula for will simplify the argument). For this specific case a simple change of variables (and maybe a little geometric argument) gives the conclusion.