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 lim_{z→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 behavior of holomorphic functions near essential singularities is described by the WeierstrassCasorati 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.
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