# Complex Analysis: #14 Isolated Singularities

Here, except for the special point z

_{0}= 0 which is not so nice, we have a good example of an analytic function. Since only a single point is causing problems with this function, let us more or less ignore this point and call it an

*isolated singularity*. So in general, an isolated singularity is a point z

_{0}∈ ℂ such that z

_{0}∉ G, yet there exists some r > 0 with B(z

_{0}, r) \ {z

_{0}} ⊂ G.

**Deﬁnition 11**

Let f : G → ℂ be analytic, but with an isolated singularity at z

_{0}∈ ℂ. The

*residue*of f at z

_{0}is the number

According to Cauchy’s theorem, the residue is a well-defined number, independent of the radius r of the circle of the path used to define it, as long as r is small enough to satisfy our condition.

We identify three different kinds of isolated singularities:

- removable singularities,
- poles,
- essential singularities.

_{0}. This singularity is

*removable*if it is possible to ﬁnd some number w

_{0}such that if we simply define f

_{0}: G ∪ {z

_{0}} → ℂ by

then f

_{0}is analytic (thus also differentiable at the special point z

_{0}). We see then that a removable singularity is really nothing special. We have simply “forgotten” to put in the correct value of the function f at the isolated point z

_{0}. By putting in the correct value, the singularity disappears.

A

*pole*is somewhat more interesting. For example the function

has a “simple” pole at the point z

_{0}= 0. But then if we multiply f with the “simple” polynomial g(z) = z, then we get f(z) · g(z) = 1. Of course the function f · g has a removable singularity at the special point z

_{0}= 0. Furthermore, the function which results does not have a zero at the point 0. The general rule is: let z

_{0}be an isolated, not removable, singularity of the analytic function f. If there is some n ∈ ℕ such that the function given by f(z) · (z − z

_{0})

^{n}has a removable singularity at z

_{0}, and the resulting function does not have a zero at z

_{0}, then z

_{0}is a pole of order n.

Finally, an

*essential*singularity is neither removable, nor is it a pole.

**Deﬁnition 12**

Let f : G → ℂ be analytic, such that all it’s isolated singularities are either removable or else they are poles. Then f is called a

*meromorphic*function (defined in G).

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