This tag is for questions relating to singularity, which is a point where a mathematical concept is not defined or well behaved, such as boundedness, differentiability, continuity. In general, because a function behaves in an anomalous manner at singular points, singularities must be treated separately when analyzing the function, or mathematical model, in which they appear.
A singularity or, singular point is a point at which a function, equation, surface, etc., becomes degenerate or just diverges towards infinity.
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.
Why we study about singularity:
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
Singularity in Complex Analysis:
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.
Classification:
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.
References:
https://en.wikipedia.org/wiki/Singularity_(mathematics)
