Singularity (mathematics)

singularitiessingularitysingularsingular pointssingular pointmathematical singularityessential singular pointsfinite-time singularitymathematical singularitiesnon-singular
In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability.wikipedia
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Algebraic curve

rational curveplane algebraic curvealgebraic curves
The algebraic curve defined by in the
This point is the only singular point of the curve.

N-vector

n''-vectorEllipsoid normal vectorgeodetic normal
A different coordinate system would eliminate the apparent discontinuity, e.g. by replacing the latitude/longitude representation with an n-vector representation.
The n-vector representation (also called geodetic normal or ellipsoid normal vector) is a three-parameter non-singular representation well-suited for replacing latitude and longitude as horizontal position representation in mathematical calculations and computer algorithms.

Essential singularity

essential singularitiessingularity
In complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits odd behavior.

Zeros and poles

polepoleszero
Thus a pole is a certain type of singularity of a function, nearby which the function behaves relatively regularly, in contrast to essential singularities, such as 0 for the logarithm function, and branch points, such as 0 for the complex square root function.

Analytic continuation

analytically continuedmeromorphic continuationanalytic extension
They may alternatively have to do with the presence of singularities.

Hyperbolic growth

hyperbolic
Mathematically the simplest finite-time singularities are power laws for various exponents, of which the simplest is hyperbolic growth, where the exponent is (negative) 1: x^{-1}.
When a quantity grows towards a singularity under a finite variation (a "finite-time singularity") it is said to undergo hyperbolic growth.

Singularity theory

singularitiessingularitysingular
See Singularity theory for singularities in differential geometry.
The theory mentioned above does not directly relate to the concept of mathematical singularity as a value at which a function is not defined.

Laurent series

Laurent expansion theoremLaurent power seriesfield of Laurent series
Laurent series with complex coefficients are an important tool in complex analysis, especially to investigate the behavior of functions near singularities.

Algebraic geometry

complex algebraic geometryalgebraiccomputational algebraic geometry
. See Singular point of an algebraic variety for details on singularities in algebraic geometry.
Since analytic varieties may have singular points, not all analytic varieties are manifolds.

Classification of discontinuities

discontinuitiesdiscontinuousdiscontinuity
In real analysis singularities are either discontinuities or discontinuities of the derivative (sometimes also discontinuities of higher order derivatives).

Euler's Disk

Other examples of finite-time singularities include the Painlevé paradox in various forms (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).
In fact, the precession rate of the axis of symmetry approaches a finite-time singularity modeled by a power law with exponent approximately −1/3 (depending on specific conditions).

Singular solution

A singular solution y s (x) of an ordinary differential equation is a solution that is singular or one for which the initial value problem (also called the Cauchy problem by some authors) fails to have a unique solution at some point on the solution.

Coin

coinsspecieexergue
Other examples of finite-time singularities include the Painlevé paradox in various forms (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).
This is mathematically modeled as a finite-time singularity – the precession rate is accelerating to infinity, before it suddenly stops, and has been studied using high speed photography and devices such as Euler's Disk.

Undefined (mathematics)

undefinedDefined and undefineddefined
In complex analysis, a point where a holomorphic function is undefined is called a singularity.

Mathematics

mathematicalmathmathematician
In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability.

Set (mathematics)

setsetsmathematical set
In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability.

Pathological (mathematics)

well-behavedpathologicalbadly behaved
In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability.

Derivative

differentiationdifferentiablefirst derivative
In real analysis singularities are either discontinuities or discontinuities of the derivative (sometimes also discontinuities of higher order derivatives). In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability.

Function of a real variable

real functionreal variablefunctions of a real variable
For example, the real function

Absolute value

modulusabsolutemagnitude
(see absolute value) also has a singularity at

Differentiable function

differentiablecontinuously differentiabledifferentiability
, since it is not differentiable there.

Cusp (singularity)

cuspcuspscusped
coordinate system has a singularity (called a cusp) at

Singular point of an algebraic variety

non-singularsmoothsingular point
. See Singular point of an algebraic variety for details on singularities in algebraic geometry.

Differential geometry

differentialdifferential geometerdifferential geometry and topology
See Singularity theory for singularities in differential geometry.

Real analysis

realtheory of functions of a real variablefunction theory
In real analysis singularities are either discontinuities or discontinuities of the derivative (sometimes also discontinuities of higher order derivatives).