A singularity is a point where a function of the complex variable $z$ stops being analytic. This means you cannot express it as an infinite series of powers of $z$ at that specific spot. But here is the catch: the function might still be analytic at points arbitrarily close to it. When this happens, we call it an isolated singularity. Because functions behave oddly at these points, you have to treat them separately when you are analyzing a mathematical model.
Let’s look at a concrete example. Take the function $f(z) = e^z / z$. This function is analytic everywhere in the complex plane. It works for all values of $z$ except one: $z = 0$. At zero, the series expansion breaks down because it includes the term $1/z$.
The series looks like this:
$$1/z + 1 + z/2 + z^2/6 + \dots + z^n/(n+1)! + \dots$$
The factorial symbol ($k!$) just means the product of integers from $k$ down to 1. The presence of $1/z$ makes the expansion undefined at the origin.
Not all singularities are created equal. How you handle them depends on the function’s behavior nearby. If the function stays bounded in a neighborhood around the singularity, you can redefine it at that point to remove the problem. This is known as a removable singularity.
But what if the function doesn’t stay bounded? Consider our earlier example again. As $z$ approaches 0, $e^z / z$ shoots off to infinity. It is not bounded. Therefore, the singularity at $z = 0$ is not removable. In this specific case, we call it a simple pole.
When you are analyzing complex functions, identifying which type of singularity you are dealing with is the first step. It determines whether you can fix the issue by redefining the function or if you have to accept the pole as an inherent part of the model.
