Definitions

# 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) = e1/z has an essential singularity at z = 0.

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

$lim_\left\{z to a\right\}f\left(z\right)$
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

• Essential Singularity at Mathworld. Retrieved on 18 February., 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