Singularity (mathematics)

, where the value of the function is not defined, as involving a division by zero.

coordinate system has a singularity (called a cusp) at

To describe the way these two types of limits are being used, suppose that

from above, regardless of the actual value the function has at the point where

The limits in this case are not infinite, but rather undefined: there is no value that

Borrowing from complex analysis, this is sometimes called an essential singularity.

In real analysis, a singularity or discontinuity is a property of a function alone.

A coordinate singularity occurs 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 in spherical 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.

In complex analysis, there are several classes of singularities.

is a function that is complex differentiable in the complement of a point

These are termed nonisolated singularities, of which there are two types: Branch points are generally the result of a multi-valued function, such as

The cut is a line or curve excluded from the domain to introduce a technical separation between discontinuous values of the function.

A finite-time singularity occurs when one input variable is time, and an output variable increases towards infinity at a finite time.

These are important in kinematics and Partial Differential Equations – infinites do not occur physically, but the behavior near the singularity is often of interest.

Mathematically, the simplest finite-time singularities are power laws for various exponents of the form

of which the simplest is hyperbolic growth, where the exponent is (negative) 1:

More precisely, in order to get a singularity at positive time as time advances (so the output grows to infinity), one instead uses

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 the various forms of the Painlevé paradox (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).

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

In algebraic geometry, a singularity of an algebraic variety is a point of the variety where the tangent space may not be regularly defined.

The simplest example of singularities are curves that cross themselves.

For example, the equation y2 − x3 = 0 defines a curve that has a cusp at the origin x = y = 0.

In fact, in this case, the x-axis is a "double tangent."

For affine and projective varieties, the singularities are the points where the Jacobian matrix has a rank which is lower than at other points of the variety.

An equivalent definition in terms of commutative algebra may be given, which extends to abstract varieties and schemes: A point is singular if the local ring at this point is not a regular local ring.