In complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits extreme behavior.
Basically, the category essential singularity is a "leftover" or default group of singularities that are especially unmanageable: by definition they fit into neither of the other two categories of singularity that may be dealt with in some manner – removable singularities and poles.
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Formal description
Consider an open subset U of the complex plane C. Let a be an element of U, and f : U \ {a} → C a meromorphic function. The point a is called an essential singularity of the function f if the singularity is neither a pole nor a removable singularity.
For example, the function f(z) = e^{1/z} has an essential singularity at z = 0.
Alternate descriptions
Let a be a complex number, assume that f(z) is not defined at a but is analytic in some region U of the complex plane, and that every open neighbourhood of a has nonempty intersection with U.
If both
If
Similarly, if
If neither
Another way to characterize an essential singularity is that the Laurent series of f at the point a has infinitely many negative degree terms (i.e., the principal part of the Laurent series 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 many times.
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