Singularity (mathematics) – Wikipedia

Point where a function, a curve or another mathematical object does not behave regularly

In mathematics, a singularity is 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 by lacking differentiability or analyticity.[1][2][3]

For example, the real function

has a singularity at x = 0 {displaystyle x=0} , where the numerical value of the function approaches {displaystyle pm infty } so the function is not defined. The absolute value function g ( x ) = | x | {displaystyle g(x)=|x|} also has a singularity at x = 0 {displaystyle x=0} , since it is not differentiable there.[4]

The algebraic curve defined by { ( x , y ) : y 3 x 2 = 0 } {displaystyle left{(x,y):y^{3}-x^{2}=0right}} in the ( x , y ) {displaystyle (x,y)} coordinate system has a singularity (called a cusp) at ( 0 , 0 ) {displaystyle (0,0)} . For singularities in algebraic geometry, see singular point of an algebraic variety. For singularities in differential geometry, see singularity theory.

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: typeI, which has two subtypes, and typeII, 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 f ( x ) {displaystyle f(x)} is a function of a real argument x {displaystyle x} , and for any value of its argument, say c {displaystyle c} , then the left-handed limit, f ( c ) {displaystyle f(c^{-})} , and the right-handed limit, f ( c + ) {displaystyle f(c^{+})} , are defined by:

The value f ( c ) {displaystyle f(c^{-})} is the value that the function f ( x ) {displaystyle f(x)} tends towards as the value x {displaystyle x} approaches c {displaystyle c} from below, and the value f ( c + ) {displaystyle f(c^{+})} is the value that the function f ( x ) {displaystyle f(x)} tends towards as the value x {displaystyle x} approaches c {displaystyle c} from above, regardless of the actual value the function has at the point where x = c {displaystyle x=c} .

There are some functions for which these limits do not exist at all. For example, the function

does not tend towards anything as x {displaystyle x} approaches c = 0 {displaystyle c=0} . The limits in this case are not infinite, but rather undefined: there is no value that g ( x ) {displaystyle g(x)} settles in on. Borrowing from complex analysis, this is sometimes called an essential singularity.

The possible cases at a given value c {displaystyle c} 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.

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 (for example, along the line 0 degrees longitude) on the surface of a sphere will suddenly experience an instantaneous change in longitude at the pole (in the case of the example, jumping from longitude 0 to longitude 180 degrees). 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 (e.g., by replacing the latitude/longitude representation with an n-vector representation).

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

Suppose that f {displaystyle f} is a function that is complex differentiable in the complement of a point a {displaystyle a} in an open subset U {displaystyle U} of the complex numbers C . {displaystyle mathbb {C} .} 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:

Branch points are generally the result of a multi-valued function, such as z {displaystyle {sqrt {z}}} or log ( z ) {displaystyle log(z)} , 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 (such as z = 0 {displaystyle z=0} and z = {displaystyle z=infty } for log ( z ) {displaystyle log(z)} ) which are fixed in place.

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 x , {displaystyle x^{-alpha },} of which the simplest is hyperbolic growth, where the exponent is (negative) 1: x 1 . {displaystyle x^{-1}.} More precisely, in order to get a singularity at positive time as time advances (so the output grows to infinity), one instead uses ( t 0 t ) {displaystyle (t_{0}-t)^{-alpha }} (using t for time, reversing direction to t {displaystyle -t} so that time increases to infinity, and shifting the singularity forward from 0 to a fixed time t 0 {displaystyle t_{0}} ).

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 Painlev 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 infinitebefore 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).

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 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.

See more here:

Singularity (mathematics) - Wikipedia