You must have heard the term “singularity” being used in various contexts. We see it in mathematics, physics, circuit design, mechanics, fluid dynamics, etc. The word “singular” means something that is extraordinary, unique, and strange. When we talk about singularity in mathematics, we usually refer to the uniqueness of mathematical objects. In particular, singularities refer to the points where the mathematical objects are not well-behaved i.e. we can’t define them for those points. In higher mathematics, we usually define functions to explain the behavior of any system. These functions are the mathematical objects under consideration, and we will see how to understand their singularities. But why do we care about singularities? Is it just a mathematical concept or do we ever see it in real life?
Why do we care about this?
The study of singularity is extremely important in many different fields. We employ complex mathematical formulations when we build physical structures and surfaces. These formulations are governed by the underlying functions, and if we don’t understand the singularities of those functions, the physical structure will collapse. Apart from this, they are used in particle physics, quantum mechanics, relativity, study of deformable surfaces, light patterns, and many more fields. We construct so many devices based on these physical phenomena, and all of them are critically dependent on their corresponding singularities.
What exactly is a singularity?
A singularity is a point at which a function, equation, surface, etc., becomes degenerate or just diverges towards infinity. Now what does “degenerate” mean? When we say something is degenerate, it means that it is the limiting case in which a class of objects changes its nature. Once it changes its nature, it usually belongs to another simpler class.
For example, let’s consider the class of rectangles. You can construct infinitely many rectangles of different sizes. But once you make the length equal to its breadth, it no longer remains a rectangle. It actually becomes a square! Here, we say that the square is a degenerate case of a rectangle. As we can see here, the class changed its nature and now it belongs to a simpler class i.e. the class of squares. Other examples include circle and a point. The point is a degenerate case of the circle as the radius approaches 0. The circle, in turn, is a degenerate form of an ellipse as the eccentricity approaches 0.
Let’s consider quadratic equations for a moment. We all know that every quadratic equation is supposed to have two roots. The roots are unique in most cases, except when the equation looks something like this: (x-3)^2 = 0. Here, we can see that both the roots are equal to 3. Since the n roots of an nth degree polynomial are usually distinct, roots which coincide are said to be degenerate. Degenerate cases often require special treatment in numerical and analytical solutions.
A Simple Example
Let’s consider the following function:
f(x) = 1/x
This function is well behaved at all points except at x=0. When you put x=0, the function is not defined because it explodes to infinity. Hence we can say that the function has a singularity at x=0.
Singularity is not just limited to the function, but also its derivatives. The derivative refers to the rate of change of that function. Let’s consider the parabola given by the equation:
y^2 = x - 2
It basically looks like a C-shaped curve with the tip at x=2. We can see that the tangent to the curve at x=2 is vertical. What it means is that the derivative of this function at this point diverges to infinity. Hence, we can say that there is a singularity at x=2.
It’s just one point out of possibly millions of points. Do I really need to worry about these singularities?
Singularities are extremely important in complex analysis, as they characterize the possible behaviors of analytic functions. Complex analysis refers to analysis of functions whose domain and range can include the complex number set. Complex singularities are basically points in the domain of a function where it fails to be analytic.
In real life, when we use one of these functions to build something real and tangible, we cannot afford to leave these singularities alone. It’s like saying I will construct this amazing bridge and it will work for every load except for 9000 pounds. When the load is exactly 9000 pounds, it will collapse and everyone on the bridge will be crushed under it. Now who would want to get anywhere near that bridge? So we need to understand if we can take care of these singularities governing the underlying functions.
How do we take care of these singularities?
Singularities can be non-isolated or isolated. Non-isolated singularities usually arise due to our own definitions of boundaries, like if we choose to define the function only within a certain limit. They are not very interesting to us because we know exactly why they occur. Isolated singularities, on the other hand, arise due the inherent nature of the functions. They refer to those isolated points where the function behavior is not defined. Isolated singularities may be classified as removable singularities, poles, essential singularities, and logarithmic singularities. The actual mathematical definitions of these terms are too involved, so I will cut through the jargon and keep it simple.
To demonstrate, I will take a really simple example. let’s consider the following function:
f(z) = (z³ - 2z)/z
If you put z=0, this function takes the 0/0 form and becomes undefined. So we can say that the function f(z) is singular at z=0. But, we can see that f(z) is equal to z²-2 everywhere except at z=0. So, as we can see, we can just define the function ourselves at z=0 to be -2 and make sure the function is well-behaved everywhere. This is an example of removable singularity.
Now, let’s consider the following equation:
f(z) = 1/z
This function clearly diverges as z approaches 0 and there’s nothing we can do about it. This type of singularity is called “pole”. We say that the function has a pole at z=0. Analyzing the poles of a system is extremely critical in circuit design.
A point where the function neither has a pole nor a removable singularity is called essential singularity of that function. A popular example that’s usually stated here is the function e^(1/z) at z=0. I simplified the concept of essential singularity way too much, and I’m feeling a bit uneasy about it! Essential singularity is way more involved than that and requires some rigorous mathematics to explain. But then again, if I do that, it will go against the spirit of a general healthy discussion. So I will just say that essential singularity is an extreme case where the function behavior becomes unmanageable. Needless to say, people make sure they avoid these essential singularities at all costs when they design real systems.