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?