This is about the situation where the Laurent series of f has nonzero principal part. When this occurs, we say that z_{0} is a singularity. The singularities are isolated if each is the center of a ball such that f is analytic except for the center of the ball.
Definition 16.6.1 Let B^{′}
It turns out isolated singularities can be neatly classified into three types, removable singularities, poles, and essential singularities. The next theorem deals with the case of a removable singularity.
Definition 16.6.2 An isolated singularity of f is said to be removable if there exists an analytic function g analytic at a and near a such that f = g at all points near a.

Thus the above limit occurs if and only if there exists a unique analytic function, g : B
Proof: ⇒Let h

where a_{0} = a_{1} = 0 because of the observation above that h^{′}

⇐The converse is obvious. ■
What of the other case where the singularity is not removable? This situation is dealt with by the amazing Casorati Weierstrass theorem. This theorem pertains to the case where f has values in ℂ. Most of what has been presented pertains to functions which have values in X for X a Banach space but this one is a specialization.
Theorem 16.6.4 (Casorati Weierstrass) Let a be an isolated singularity and suppose for some r > 0, f
 (16.11) 
where g
Proof: Suppose B
 (16.12) 
There are two cases. First suppose h

Hence, taking both sides to the −1 power,

and so 16.11 holds.
The other case is that h

a function analytic near a. Therefore, the singularity is removable. ■
This theorem is the basis for the following definition which describes isolated singularities.
Definition 16.6.5 Let a be an isolated singularity of a X valued function f. When 16.11 holds for z near a, then a is called a pole. The order of the pole in 16.11 is M. Essential singularities are those which have infinitely many nonzero terms in the principal part of the Laurent series. When a function f is analytic except for isolated singularities and the isolated singularities are all poles, and there are finitely many of these poles in every compact set, the function is called meromorphic.
Actually, if you insist only that the singularities are isolated and poles, then you can prove that there are finitely many in any compact set so part of the above definition is actually redundant, but this will be shown later. What follows is the definition of something called a residue. This pertains to a singularity which has a pole at an isolated singularity.
Definition 16.6.6 The residue of f at an isolated singularity α which is a pole, written res

Thus res