# 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).