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 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 :$lim\_\{z\; o\; a\}f(z)$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.

References

*citeweb|url=http://mathworld.wolfram.com/EssentialSingularity.html|title=Essential Singularity at Mathworld|accessdate=18 February|accessyear=2008
*Lars V. Ahlfors; "Complex Analysis", McGraw-Hill, 1979
*Rajendra Kumar Jain, S. R. K. Iyengar; "Advanced Engineering Mathematics". Page 920. Alpha Science International, Limited, 2004. ISBN 1842651854

External links

* " [http://demonstrations.wolfram.com/AnEssentialSingularity/ An Essential Singularity] " by Stephen Wolfram, The Wolfram Demonstrations Project.
* [http://planetmath.org/encyclopedia/EssentialSingularity.html Essential Singularity on Planet Math]