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. See singularity theory for general discussion of the geometric theory, which only covers some aspects.

For example, the function

f(x)=\frac{1}{x}

on the real line has a singularity at x = 0, where it seems to "explode" to ±? and is not defined. The function g(x) = |x| (see absolute value) also has a singularity at x = 0, since it is not differentiable there. Similarly, the graph defined by y² = x also has a singularity at (0,0), this time because it has a "corner" (vertical tangent) at that point.

The algebraic set defined by y² = x² in the (x, y) coordinate system has a singularity (singular point) at (0, 0) because it does not admit a tangent there.

Real analysis

In real analysis singularities are also called discontinuities. There are three kinds: type I, which has two sub-types, and type II.

In order to describe these types, suppose that f(x) is a function of a real argument x, and for any value of its argument, say c, the symbols f(c^+) and 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 limit f(c^-) is called the left-handed limit, and f(c^+) is called the right-handed limit. The value f(c^-) is the value that the function f(x) tends towards as the value x approaches c from below, and the value f(c^+) is the value that the function f(x) tends towards as the value x approaches c from above, 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(\frac{1}{x})

does not tend towards anything as x approaches c = 0. The limits in this case are not infinite, but rather undefined: there is no value that g(x) settles in on.

• A point of continuity, which is not a singularity, is a value of c for which f(c^-) = f(c) = f(c^+), as one usually expects. All the values must be finite.
• A type I discontinuity occurs when both f(c^-) and f(c^+) both exist (and are both finite), but either f(c^-) \neq f(c^+) or f(c) either does not exist for that value of x, or does not match the value that the two limits tend towards. Two subtypes occur:
  • A removable discontinuity occurs when f(c^-) = f(c^+), but either the value of f(c) does not match the limits, or the function does not exist at the point x = c .
  • A jump discontinuity occurs when f(c^-) \neq f(c^+), regardless of whether f(c) exists, and regardless of what value it might have if it does exist.
• A type II discontinuity occurs when either f(c^-) or f(c^+) does not exist (possibly both). This has one subtype:
  • 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 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.

Complex analysis

In complex analysis, there are four kinds of singularity, to be described below. Suppose U is an open subset of the complex numbers C, a is an element of U, and f is a holomorphic function defined on U \ {a}.

• Isolated singularities:Suppose f is not defined at a, although defined on U \ {a}.
  • The point a is a removable singularity of f if there exists a holomorphic function g defined on all of U such that f(z) = g(z) for all z in U − {a}. The function g replaces the function f in all but name.
  • The point a is a pole or non-essential singularity of f if there exists a holomorphic function g defined on U and a natural number n such that f(z) = g(z) / (z − a)ⁿ for all z in U − {a}. The derivative at a non-essential singularity may or may not exist. If g(a) is nonzero, then we say that a is a pole of order n.
  • The point a is an essential singularity of f if is neither a removable singularity nor a pole. The point a is an essential singularity if and only if the Laurent series has infinitely many powers of negative degree.
• Branch points. In short, the branch points are generally the result of a multi-valued function, such as \sqrt{z} being defined within a certain interval so that it behaves like a single-valued function. The function may have different values on each side of the branch cut so every point along the branch cut has no derivative.

From the point of view of dynamics

A finite-time singularity occurs when a kinematic variable increases towards infinity at a finite time. 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 Euler's disk, the Painlevé paradox, and Heinz von Foerster's Doomsday's Equation.

Algebraic geometry and commutative algebra

In algebraic geometry and commutative algebra, a singularity is a prime ideal whose localization is not a regular local ring (alternately a scheme with a stalk that is not a regular local ring). For example, y^2 - x^3 = 0 defines an isolated singular point (at the cusp) x = y = 0. The ring in question is given by

\mathbb{C}[x,y] / (y^2 - x^3) \cong \mathbb{C}[t^2, t^3].

The maximal ideal of the localization at (t^2, t^3) is a height one local ring generated by two elements and thus not regular.

Singular matrices

In linear algebra a square matrix is said to be singular when it is not invertible, that is when its determinant is zero.

Singular value decomposition

Singular value decomposition (SVD) is a method of factorizing matrices. A non-negative real number ? is a singular value for M if and only if there exist normalized vectors u in K^m and v in Kⁿ such that

Mv = \sigma u \,\mbox{ and } M^*u = \sigma v. \,\!

The vectors u and v are called left-singular and right-singular vectors for ?, respectively. The factorisation is

M = U\Sigma V^* \,\!

where diagonal entries of ? are equal to the singular values of M. The columns of U and V are left- resp. right-singular vectors for the corresponding singular values. It is widely used in statistics where it is used as a technique for solving linear least squares problems and is related to principal components analysis.

See also

• Asymptote
• Catastrophe theory
• Discontinuity
• Defined and undefined
• Infinity
• Singular solution of a differential equation

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