Essential singularity

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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.

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

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

The point a is an essential singularity if and only if the limit

does not exist as a complex number nor equals infinity. This is the case if and only if either f has poles in every neighbourhood of a or the Laurent series of f at the point a has infinitely many negative degree terms (i.e. the principal part 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

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