Singularity (mathematics)

Inmathematics,asingularityis a point at which a given mathematical object is not defined, or a point where the mathematical object ceases to bewell-behavedin some particular way, such as by lackingdifferentiabilityoranalyticity.

For example, thereciprocal function$f(x)=1/x$has a singularity at$x=0$,where the value of thefunctionis not defined, as involving adivision by zero.Theabsolute valuefunction$g(x)=|x|$also has a singularity at$x=0$,since it is notdifferentiablethere.

Thealgebraic curvedefined by$\left\{(x,y):y^{3}-x^{2}=0\right\}$in the$(x,y)$coordinate system has a singularity (called acusp) at$(0,0)$.For singularities inalgebraic geometry,seesingular point of an algebraic variety.For singularities indifferential geometry,seesingularity theory.

Real analysis

Inreal analysis,singularities are eitherdiscontinuities,or discontinuities of thederivative(sometimes also discontinuities of higher order derivatives). There are four kinds of discontinuities:type I,which has two subtypes, andtype 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$f(x)$is a function of a real argument$x$,and for any value of its argument, say$c$,then theleft-handed limit,$f(c^{-})$,and theright-handed limit,$f(c^{+})$,are defined by:

$f(c^{-})=\lim _{x\to c}f(x)$,constrained by$x<c$and

$f(c^{+})=\lim _{x\to c}f(x)$,constrained by$x>c$.

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

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

$g(x)=\sin \left({\frac {1}{x}}\right)$

does not tend towards anything as$x$approaches$c=0$.The limits in this case are not infinite, but ratherundefined:there is no value that$g(x)$settles in on. Borrowing from complex analysis, this is sometimes called anessential singularity.

The possible cases at a given value$c$for the argument are as follows.

• Apoint of continuityis a value of$c$for which$f(c^{-})=f(c)=f(c^{+})$,as one expects for a smooth function. All the values must be finite. If$c$is not a point of continuity, then a discontinuity occurs at$c$.
• Atype Idiscontinuity occurs when both$f(c^{-})$and$f(c^{+})$exist and are finite, but at least one of the following three conditions also applies:
  • $f(c^{-})\neq f(c^{+})$;
  • $f(x)$is not defined for the case of$x=c$;or
  • $f(c)$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:

  • Ajump discontinuityoccurs when$f(c^{-})\neq f(c^{+})$,regardless of whether$f(c)$is defined, and regardless of its value if it is defined.
  • Aremovable discontinuityoccurs when$f(c^{-})=f(c^{+})$,also regardless of whether$f(c)$is defined, and regardless of its value if it is defined (but which does not match that of the two limits).
• Atype IIdiscontinuity occurs when either$f(c^{-})$or$f(c^{+})$does not exist (possibly both). This has two subtypes, which are usually not considered separately:
  • Aninfinite discontinuityis 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 itsgraphhas avertical asymptote.
  • Anessential singularityis a term borrowed from complex analysis (see below). This is the case when either one or the other limits$f(c^{-})$or$f(c^{+})$does not exist, but not because it is aninfinite discontinuity.Essential singularitiesapproach no limit, not even if valid answers are extended to include$\pm \infty$.

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

Main article:Coordinate singularity

Acoordinate singularityoccurs 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 inspherical 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 ann-vectorrepresentation).

Complex analysis

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

Isolated singularities

Suppose that$f$is a function that iscomplex differentiablein thecomplementof a point$a$in anopen subset$U$of thecomplex numbers$\mathbb {C}.$Then:

• The point$a$is aremovable singularityof$f$if there exists aholomorphic function$g$defined on all of$U$such that$f(z)=g(z)$for all$z$in$U\smallsetminus \{a\}.$The function$g$is a continuous replacement for the function$f.$
• The point$a$is apoleor non-essential singularity of$f$if there exists a holomorphic function$g$defined on$U$with$g(a)$nonzero, and anatural number$n$such that$f(z)={\frac {g(z)}{(z-a)^{n}}}$for all$z$in$U\smallsetminus \{a\}.$The least such number$n$is called theorder of the pole.The derivative at a non-essential singularity itself has a non-essential singularity, with$n$increased by1(except if$n$is0so that the singularity is removable).
• The point$a$is anessential singularityof$f$if it is neither a removable singularity nor a pole. The point$a$is an essential singularityif and only iftheLaurent serieshas infinitely many powers of negative degree.

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 pointsof isolated singularities. If they are all poles, despite admittingLaurent 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 beanalytically continuedaround (or outside them if they are closed curves in theRiemann sphere).

Branch points

Branch pointsare generally the result of amulti-valued function,such as${\sqrt {z}}$or$\log(z),$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$z=0$and$z=\infty$for$\log(z)$) which are fixed in place.

Finite-time singularity

Thereciprocal function,exhibitinghyperbolic growth.

Afinite-time singularityoccurs when one input variable is time, and an output variable increases towards infinity at a finite time. These are important inkinematicsandPartial Differential Equations– infinites do not occur physically, but the behavior near the singularity is often of interest. Mathematically, the simplest finite-time singularities arepower lawsfor various exponents of the form$x^{-\alpha },$of which the simplest ishyperbolic growth,where the exponent is (negative) 1:$x^{-1}.$More precisely, in order to get a singularity at positive time as time advances (so the output grows to infinity), one instead uses$(t_{0}-t)^{-\alpha }$(usingtfor time, reversing direction to$-t$so that time increases to infinity, and shifting the singularity forward from 0 to a fixed time$t_{0}$).

An example would be the bouncing motion of an inelastic ball on a plane. If idealized motion is considered, in which the same fraction ofkinetic energyis lost on each bounce, thefrequencyof bounces becomes infinite, as the ball comes to rest in a finite time. Other examples of finite-time singularities include the various forms of thePainlevé paradox(for example, the tendency of a chalk to skip when dragged across a blackboard), and how theprecessionrate of acoinspun on a flat surface accelerates towards infinite—before abruptly stopping (as studied using theEuler's Disktoy).

Hypothetical examples includeHeinz von Foerster's facetious "Doomsday's equation"(simplistic models yield infinite human population in finite time).

Algebraic geometry and commutative algebra

Inalgebraic geometry,asingularity of an algebraic varietyis a point of the variety where thetangent spacemay not be regularly defined. The simplest example of singularities are curves that cross themselves. But there are other types of singularities, likecusps.For example, the equationy²−x³= 0defines a curve that has a cusp at the originx=y= 0.One could define thex-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, thex-axis is a "double tangent."

Foraffineandprojective varieties,the singularities are the points where theJacobian matrixhas arankwhich is lower than at other points of the variety.

An equivalent definition in terms ofcommutative algebramay be given, which extends toabstract varietiesandschemes:A point issingularif thelocal ring at this pointis not aregular local ring.

See also

• Catastrophe theory
• Defined and undefined
• Degeneracy (mathematics)
• Hyperbolic growth
• Movable singularity
• Pathological (mathematics)
• Regular singularity
• Singular solution