Singularity (mathematics)
In
For example, the
The algebraic curve defined by in the coordinate system has a singularity (called a cusp) 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 (sometimes also discontinuities of higher order derivatives). There are four kinds of discontinuities: type I, which has two subtypes, and type II, which can also be divided into two subtypes (though usually is not).
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:
- , constrained by and
- , constrained 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.
- A point of continuity is a value of for which , as one expects for a smooth function. All the values must be finite. If is not a point of continuity, then a discontinuity occurs at .
- A type I discontinuity occurs when both and exist and are finite, but at least one of the following three conditions also applies:
- ;
- is not defined for the case of ; or
- has a defined value, which, however, does not match the value of the two limits.
- Type I discontinuities can be further distinguished as being one of the following subtypes:
- A jump discontinuityoccurs when , regardless of whether is defined, and regardless of its value if it is defined.
- A removable discontinuity occurs when , also regardless of whether is defined, and regardless of its value if it is defined (but which does not match that of the two limits).
- A type II discontinuity occurs when either or does not exist (possibly both). This has two subtypes, which are usually not considered separately:
- An infinite discontinuity is the special case when either the left hand or right hand limit does not exist, specifically because it is infinite, and the other limit is either also infinite, or is some well defined finite number. In other words, the function has an infinite discontinuity when its vertical asymptote.
- An essential singularity is a term borrowed from complex analysis (see below). This is the case when either one or the other limits or does not exist, but not because it is an infinite discontinuity. Essential singularities approach no limit, not even if valid answers are extended to include .
- An infinite discontinuity is the special case when either the left hand or right hand limit does not exist, specifically because it is infinite, and the other limit is either also infinite, or is some well defined finite number. In other words, the function has an infinite discontinuity when its
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
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 is a function that is
- The point is a removable singularity of if there exists a holomorphic function defined on all of such that for all in The function is a continuous replacement for the function [5]
- The point is a poleor non-essential singularity of if there exists a holomorphic function defined on with nonzero, and a natural number such that for all in The least such number is called the order of the pole. The derivative at a non-essential singularity itself has a non-essential singularity, with increased by 1 (except if is 0 so that the singularity is removable).
- The point is an essential singularity of if it is neither a removable singularity nor a pole. The point is an essential singularity if and only if the Laurent series has infinitely many powers of negative degree.[1]
Nonisolated singularities
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:
- Cluster points: limit points of isolated singularities. If they are all poles, despite admitting Laurent seriesexpansions on each of them, then no such expansion is possible at its limit.
- Natural boundaries: any non-isolated set (e.g. a curve) on which functions cannot be analytically continued around (or outside them if they are closed curves in the Riemann sphere).
Branch points
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
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 (for example, the tendency of a chalk to skip when dragged across a blackboard), and how the precession rate of a coin spun on a flat surface accelerates towards infinite—before abruptly stopping (as studied using the Euler's Disk toy).
Hypothetical examples include Heinz von Foerster's facetious "Doomsday's equation" (simplistic models yield infinite human population in finite time).
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 y2 − x3 = 0 defines a curve that has a cusp at the origin x = y = 0. One could define the x-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 x-axis is a "double tangent."
For
An equivalent definition in terms of
See also
- Catastrophe theory
- Defined and undefined
- Degeneracy (mathematics)
- Hyperbolic growth
- Pathological (mathematics)
- Singular solution
- Removable singularity
References
- ^ a b "Singularities, Zeros, and Poles". mathfaculty.fullerton.edu. Retrieved 2019-12-12.
- ^ "Singularity | complex functions". Encyclopedia Britannica. Retrieved 2019-12-12.
- ^ "Singularity (mathematics)". TheFreeDictionary.com. Retrieved 2019-12-12.
- ISBN 978-1-305-46505-3.
- ^ Weisstein, Eric W. "Singularity". mathworld.wolfram.com. Retrieved 2019-12-12.