Home > 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 isn't 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.
1 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 numberNatural number can mean either a positive integer ( 1, 2, 3, 4,. or a non-negative integer ( 0, 1, 2, 3, 4,. Natural numbers have two main purposes: they can be used for counting ("there are 3 apples on the table"), or they can be used for ordering ("this n such that f(z) = g(z) / (z − a)n for all z in U − {a}.
- The point a is an essential singularityIn complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits extreme behavior. Formally, consider an open subset U of the complex plane C an element a of U and a holomorphic function f defined on U of f if it is neither a removable singularity nor a pole.
- A branch pointIn complex analysis, a branch point may be thought of informally as a point z at which a " multiple-valued function" changes values when one winds once around z''. Examples: 0 is a branch point of the square root function. Suppose w &radic z and z starts of f is one requiring a more verbose definition; see the article of that title.
2 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 energyKinetic energy (also called vis viva or living force is energy possessed by a body by virtue of its motion. The kinetic energy of a body is equal to the amount of work needed to establish its velocity and rotation, starting from rest. Equations Definition is lost on each bounce, the frequencyFrequency is the measurement of the number of times that a repeated event occurs per unit time. To calculate the frequency, one fixes a time interval, counts the number of occurrences of the event within that interval, and then divides this count by the l of bounces becomes infinite as the ball comes to rest in a finite time. Other examples of finite-time singularities include Euler's diskEuler's disk refers to a circular disk that spins, without slipping, on a surface. The canonical example is a coin spinning on a table; the concept is named after Leonard Euler. It is universally observed that a spinning Euler's disk ultimately comes to r and the Painlevé paradox .
Mathematical analysisAnalysis is that branch of mathematics which deals with the real numbers and complex numbers and their functions. It has its beginnings in the rigorous formulation of calculus and studies concepts such as continuity, integration and differentiability in g
