Essential singularity

From Wikipedia, the free encyclopedia

Plot of the function exp(1/z), centered on the essential singularity at z=0. The hue represents the complex argument, the luminance represents the absolute value. This plot shows how approaching the essential singularity from different directions yields different behaviors (as opposed to a pole, which would be uniformly white).

In complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits extreme behavior.

Basically, the category essential singularity is a "left-over" or default group of singularities that are especially unmanageable: by definition they fit into neither of the other two categories of singularity that be dealt with in some manner – removable singularities and poles (which see).

[edit] Formal description

Formally, consider an open subset U in the complex plane C. If there is an element a in U, and a meromorphic function f : U \ {a} → C. The point a is called an essential singularity of the function f if the singularity is neither a pole nor a removable singularity.

For example, the function f(z) = e^1/z has an essential singularity at z = 0.

[edit] Alternate descriptions

Let a be a complex number, assume that f(z) is not defined at a but is analytic in some region U of the complex plane, and that every open neighbourhood of a has non-empty intersection with U.

If both

  and     exist, then a is a removable singularity of both f and 1/f.

If

  exists but     does not exist, then a is a zero of f and a pole of 1/f.

Similarly, if

  does not exist but     does exist, then a is a pole of f and a zero of 1/f.

If neither

  nor     exists, then a is an essential singularity of both f and 1/f.

Another way to characterize an essential singularity is

The point a is an essential singularity if and only if one (or both) of these of two conditions holds:

• The function f has poles in every neighbourhood of a, meaning that the singularity is not isolated.
• The Laurent series of f at the point a has infinitely many negative degree terms (i.e. the principal part of the Laurent series is an infinite sum).

The behavior of meromorphic functions near essential singularities is described by the Weierstrass–Casorati theorem and by the considerably stronger Picard's great theorem. The latter says that in every neighborhood of an essential singularity a, the function f takes on every complex value, except possibly one, infinitely often.

[edit] References

[edit] External links

• An Essential Singularity by Stephen Wolfram, Wolfram Demonstrations Project.
• Essential Singularity on Planet Math

This mathematics-related article is a stub. You can help Wikipedia by expanding it.