Singularity (mathematics)


In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point where the mathematical object ceases to be well-behaved in some particular way, such as the lack of differentiability or analyticity.
For example, the real function
has a singularity at, where it seems to "explode" to and is hence not defined. The absolute value function also has a singularity at, since it is not differentiable there.
The algebraic curve defined by in the coordinate system has a singularity at. For singularities in algebraic geometry, see singular point of an algebraic variety. For singularities in differential geometry, see singularity theory.

Real analysis

In real analysis, singularities are either discontinuities, or discontinuities of the derivative. There are four kinds of discontinuities: type I, which has two subtypes, and type II, which can also be divided into two subtypes.
To describe the way these two types of limits are being used, suppose that is a function of a real argument, and for any value of its argument, say, then the left-handed limit,, and the right-handed limit,, are defined by:
The value is the value that the function tends towards as the value approaches from below, and the value is the value that the function tends towards as the value approaches from above, regardless of the actual value the function has at the point where .
There are some functions for which these limits do not exist at all. For example, the function
does not tend towards anything as approaches. The limits in this case are not infinite, but rather undefined: there is no value that settles in on. Borrowing from complex analysis, this is sometimes called an essential singularity.
The possible cases at a given value for the argument are as follows.
In real analysis, a singularity or discontinuity is a property of a function alone. Any singularities that may exist in the derivative of a function are considered as belonging to the derivative, not to the original function.

Coordinate singularities

A coordinate singularity occurs when an apparent singularity or discontinuity occurs in one coordinate frame, which can be removed by choosing a different frame. An example of this is the apparent singularity at the 90 degree latitude in spherical coordinates. An object moving due north on the surface of a sphere will suddenly experience an instantaneous change in longitude at the pole. This discontinuity, however, is only apparent; it is an artifact of the coordinate system chosen, which is singular at the poles. A different coordinate system would eliminate the apparent discontinuity.

Complex analysis

In complex analysis, there are several classes of singularities. These include the isolated singularities, the nonisolated singularities and the branch points.

Isolated singularities

Suppose that U is an open subset of the complex numbers C, with the point a being an element of U, and that f is a complex differentiable function defined on some neighborhood around a, excluding a: U \, then:
Other than isolated singularities, complex functions of one variable may exhibit other singular behaviour. These are termed nonisolated singularities, of which there are two types:
s are generally the result of a multi-valued function, such as or, which are defined within a certain limited domain so that the function can be made single-valued within the domain. The cut is a line or curve excluded from the domain to introduce a technical separation between discontinuous values of the function. When the cut is genuinely required, the function will have distinctly different values on each side of the branch cut. The shape of the branch cut is a matter of choice, even though it must connect two different branch points which are fixed in place.

Finite-time singularity

A finite-time singularity occurs when one input variable is time, and an output variable increases towards infinity at a finite time. These are important in kinematics and PDEs – infinites do not occur physically, but the behavior near the singularity is often of interest. Mathematically, the simplest finite-time singularities are power laws for various exponents of the form of which the simplest is hyperbolic growth, where the exponent is 1: More precisely, in order to get a singularity at positive time as time advances, one instead uses .
An example would be the bouncing motion of an inelastic ball on a plane. If idealized motion is considered, in which the same fraction of kinetic energy is lost on each bounce, the frequency of bounces becomes infinite, as the ball comes to rest in a finite time. Other examples of finite-time singularities include the various forms of the Painlevé paradox, and how the precession rate of a coin spun on a flat surface accelerates towards infinite—before abruptly stopping.
Hypothetical examples include Heinz von Foerster's facetious "Doomsday's equation".

Algebraic geometry and commutative algebra

In algebraic geometry, a singularity of an algebraic variety is a point of the variety where the tangent space may not be regularly defined. The simplest example of singularities are curves that cross themselves. But there are other types of singularities, like cusps. For example, the equation defines a curve that has a cusp at the origin. One could define the -axis as a tangent at this point, but this definition can not be the same as the definition at other points. In fact, in this case, the -axis is a "double tangent."
For affine and projective varieties, the singularities are the points where the Jacobian matrix has a rank which is lower than at other points of the variety.
An equivalent definition in terms of commutative algebra may be given, which extends to abstract varieties and schemes: A point is singular if the local ring at this point is not a regular local ring.