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

.

Thus

.

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?