# Mathematical singularity

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) = 1/x

on the real line has a singularity at x = 0, where it explodes to ±∞ and is not defined. The function g(x) = |x| (see absolute value) also has a singularity at x = 0, since it isn't 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.

## Complex analysis

In complex analysis, there are four kinds of singularity. Suppose U is an open subset of C, a is an element of U and f is a holomorphic function 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 point a is a pole of f if there exists a holomorphic function g defined on U and a natural number n such that f(z) = g(z) / (za)n for all z in U − {a}.
• A branch point of f is one requiring a more verbose definition; see the article of that title.

## 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 and the Painlevé paradox.

## 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 (mathematics) with a stalk that is not a regular local ring). For example, $\displaystyle y^2 - x^3 = 0$ defines an isolated singular point (at the cusp) $\displaystyle x = y = 0$ . The ring in question is given by

$\displaystyle C[x,y] / (y^2 - x^3) \cong C[t^2, t^3].$

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