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Singularity

In general, a singularity is a point at which an equation, surface, etc., blows up or becomes degenerate. Singularities are often also called singular points.Singularities are extremely important in complex analysis, where they characterize the possible behaviors of analytic functions. Complex singularities are points in the domain of a function where fails to be analytic. Isolated singularities may be classified as poles, essential singularities, logarithmic singularities, or removable singularities. Nonisolated singularities may arise as natural boundaries or branch cuts.Consider the second-orderordinary differential equationIf and remain finite at , then is called an ordinary point. If either or diverges as , then is called a singular point. Singular points are further classified as follows: 1. If either or diverges as but and remain finite as , then is called a regular singular point (or nonessential singularity). 2. If..

Singular point

A singular point of an algebraic curve is a point where the curve has "nasty" behavior such as a cusp or a point of self-intersection (when the underlying field is taken as the reals). More formally, a point on a curve is singular if the and partial derivatives of are both zero at the point . (If the field is not the reals or complex numbers, then the partial derivative is computed formally using the usual rules of calculus.)The following table gives some representative named curves that have various types of singular points at their origin.singularitycurveequationacnodecuspcusp curvecrunodecardioidquadruple pointquadrifoliumramphoid cuspkeratoid cusptacnodecapricornoidtriple pointtrifoliumConsider the following two examples. For the curvethe cusp at (0, 0) is a singular point. For the curve is a nonsingular point and this curve is nonsingular.Singular points are sometimes known as singularities,and vice versa...

Removable singularity

A removable singularity is a singular point of a function for which it is possible to assign a complex number in such a way that becomes analytic. A more precise way of defining a removable singularity is as a singularity of a function about which the function is bounded. For example, the point is a removable singularity in the sinc function , since this function satisfies .

Logarithmic singularity

A logarithmic singularity is a singularity of an analytic function whose main -dependent term is of order . An example is the singularity of the Bessel function of the second kindat (where is the Euler-Mascheroni constant), plotted above along the real axis, where is shown in red and the constant and logarithmic terms shown in blue.Singularities with leading term consisting of nested logarithms, e.g., , are also considered logarithmic.A logarithmic singularity is equivalent to a logarithmicbranch point.

Isolated singularity

An isolated singularity is a singularity for which there exists a (small) real number such that there are no other singularities within a neighborhood of radius centered about the singularity. Isolated singularities are also known as conic double points.The types of isolated singularities possible for cubic surfaces have been classified (Schläfli 1863, Cayley 1869, Bruce and Wall 1979) and are summarized in the following table from Fischer (1986).namesymbolnormal formCoxeter-Dynkin diagramconic double pointbiplanar double pointbiplanar double pointbiplanar double pointbiplanar double pointuniplanar double pointuniplanar double pointuniplanar double pointelliptic cone point--

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