• Complex Analysis: #14 Isolated Singularities

A “singularity” is really nothing more than a point of the complex plane where a given function is not defined. That is, if f is defined in a region G ⊂ ℂ, then any point a ∉ G can be thought of as being a singularity of the function. But this is not really what we are thinking about when we speak of singularities. As an example of what we are thinking about, consider the particular function

Here, except for the special point z₀ = 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 z₀ ∈ ℂ such that z₀ ∉ G, yet there exists some r > 0 with B(z₀, r) \ {z₀} ⊂ G.

Definition 11
Let f : G → ℂ be analytic, but with an isolated singularity at z₀ ∈ ℂ. The residue of f at z₀ 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. 

Let’s begin with removable singularities. Let f : G → ℂ have a singularity at z₀. This singularity is removable if it is possible to find some number w₀ such that if we simply define f₀ : G ∪ {z₀} → ℂ by

then f₀ is analytic (thus also differentiable at the special point z₀). 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 z₀. 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 z₀ = 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 z₀ = 0. Furthermore, the function which results does not have a zero at the point 0. The general rule is: let z₀ 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 − z₀)ⁿ has a removable singularity at z₀, and the resulting function does not have a zero at z₀, then z₀ 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).