Encyclopedia > Essential singularity

  Article Content

Essential singularity

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 holomorphic function f defined on U - {a}. 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) = exp(1/z) has an essential singularity at a = 0.

The point a is an essential singularity if and only if the limit limza f(z) does not exist as a complex number nor equals infinity. This is the case if and only if the Laurent series of f at the point a has infinitely many negative degree terms.

The behavior of holomorphic 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 (tiny) neighborhood of a, the function f takes on every complex value, except possibly one, infinitely often.



All Wikipedia text is available under the terms of the GNU Free Documentation License

 
  Search Encyclopedia

Search over one million articles, find something about almost anything!
 
 
  
  Featured Article
Charles V, Holy Roman Emperor

... Peasants' Revolt in Germany and the formation of the Lutheran Schmalkaldic League, and Charles delegated increasing responsibility for Germany to his brother Ferdinand ...

 
 
 
This page was created in 40.1 ms