When a complex function f has an isolated singularity at a point a which is neither a pole nor a removable singularity, it is said to have an essential singularity at a. This is equivalent to saying that the Laurent series of f at a contains infinitely many terms involving negative powers of (z-a), so that f (z-a)n fails to be differentiable at a for all n.

An example of a function with an essential singularity is e1/z, which has an essential singularity at the origin.

Essential singularities are occasionally referred to as "irregular singularities".

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