I must find the singularities and classify them (poles, essential or removable singularities) and find their principal part.
The function is .
My attempt :
By a first look, I notice that the function has a pole or singularity at and .
Let's examinate what happens when .
The Taylor series of in is .
Now I'm unsure of myself. I see that the first term is undetermined when and while all the other terms are undetermined only when . As a consequence I'm tempted to say that is a pole of order . (Isn't a removable singularity also?).
would be an essential singularity since there are infinite terms where is not defined, but I don't know how to justify formally.
Plus, another doubt I have is why did I find the Taylor series of in ? I could have done it for an arbitrary , or not?