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 a singularity which 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 the Laurent series of ''f'' at the point ''a'' has infinitely many negative degree terms (the principal part is an infinite sum).
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 neighborhood of an essential singularity ''a'', the function ''f'' takes on ''every'' complex value, except possibly one, infinitely often.