Singularity (original) (raw)

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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 z_0 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-order ordinary differential equation

y^('')+P(x)y^'+Q(x)y=0.

If P(x) and Q(x) remain finite at x=x_0 , then x_0 is called an ordinary point. If either P(x) or Q(x) diverges as x->x_0 , then x_0 is called a singular point. Singular points are further classified as follows:

1. If either P(x) or Q(x) diverges as x->x_0 but (x-x_0)P(x) and (x-x_0)^2Q(x) remain finite as x->x_0 , then x=x_0 is called a regular singular point (or nonessential singularity).

2. If P(x) diverges more quickly than 1/(x-x_0) , so (x-x_0)P(x) approaches infinity as x->x_0 , or Q(x) diverges more quickly than 1/(x-x_0)^2Q so that (x-x_0)^2Q(x) goes to infinity as x->x_0 , then x_0 is called an irregular singularity (or essential singularity).

A pole of order is a point z_0 of f(z) such that the Laurent series of f(z) has a_n=0 for n<-m and a_(-m)!=0 .

Essential singularities are poles of infinite order. A pole of order is a singularity z_0 of f(z) for which the function (z-z_0)^nf(z) is nonsingular and for which (z-z_0)^kf(z) is singular for k=0 , 1, ..., n-1 .

A logarithmic singularity is a singularity of an analytic function whose main -dependent term is of order O(lnz) , O(lnlnz) , etc.

Removable singularities are singularities for which it is possible to assign a complex number in such a way that f(z) becomes analytic. For example, the function f(z)=z^2/z has a removable singularity at 0, since f(z)=z everywhere but 0, and f(z) can be set equal to 0 at z=0 . Removable singularities are not poles.