Essential singularity

For essential singularities of real valued functions, see Classification of discontinuities.

In complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits extreme behavior.

The category essential singularity is a "left-over" 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) = e1/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 non-empty intersection with U.

If both

\lim_{z \to a}f(z)   and   \lim_{z \to a}\frac{1}{f(z)}   exist, then a is a removable singularity of both f and 1/f.

If

\lim_{z \to a}f(z)   exists but   \lim_{z \to a}\frac{1}{f(z)}   does not exist, then a is a zero of f and a pole of 1/f.

Similarly, if

\lim_{z \to a}f(z)   does not exist but   \lim_{z \to a}\frac{1}{f(z)}   does exist, then a is a pole of f and a zero of 1/f.

If neither

\lim_{z \to a}f(z)   nor   \lim_{z \to a}\frac{1}{f(z)}   exists, then a is an essential singularity of both f and 1/f.

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 Casorati–Weierstrass 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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