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Essential singularity

Location around which a function displays irregular behavior

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

The category essential singularity is a "left-over" or default group of isolated 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. In practice some include non-isolated singularities too; those do not have a residue.

01Formal description

Consider an open subset $U$ of the complex plane ⁠ $\mathbb {C}$ ⁠. Let $a$ be an element of ⁠ $U$ ⁠, and $f:U\smallsetminus \{a\}\to \mathbb {C}$ a holomorphic 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 ⁠ $z=0$ ⁠.

02Alternative descriptions

Let $a$ be a complex number, and 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 $\lim _{z\to a}{1}/{f(z)}$ exist, then $a$ is a removable singularity of both $f$ and ⁠ ${1}/{f}$ ⁠.
If $\lim _{z\to a}f(z)$ exists but $\lim _{z\to a}{1}/{f(z)}$ does not exist (⁠ $\lim _{z\to a}\left\vert {1}/{f(z)}\right\vert =\infty$ ⁠), then $a$ is a zero of $f$ and a pole of ⁠ ${1}/{f}$ ⁠.
If $\lim _{z\to a}f(z)$ does not exist (in fact ⁠ ${1}$ ⁠) but $\lim _{z\to a}{1}/{f(z)}$ exists, then $a$ is a pole of $f$ and a zero of ⁠ ${1}/{f}$ ⁠.
If neither $\lim _{z\to a}f(z)$ nor $\lim _{z\to a}{1}/{f(z)}$ 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). A related definition is that if there is a point $a$ for which $f(z)(z-a)^{n}$ is not differentiable for any integer ⁠ $n>0$ ⁠, then $a$ is an essential singularity of ⁠ $f$ ⁠.

On a Riemann sphere with a point at infinity, ⁠ $\infty _{\mathbb {C} }$ ⁠, the function ${f(z)}$ has an essential singularity at that point if and only if the ${f(1/z)}$ has an essential singularity at ⁠ $0$ ⁠: i.e. neither $\lim _{z\to 0}{f(1/z)}$ nor $\lim _{z\to 0}{1}/{f(1/z)}$ exists. The Riemann zeta function on the Riemann sphere has only one essential singularity, which is at ⁠ $\infty _{\mathbb {C} }$ ⁠. Indeed, every meromorphic function aside that is not a rational function has a unique essential singularity at ⁠ $\infty _{\mathbb {C} }$ ⁠.

The behavior of holomorphic functions near their 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. (The exception is necessary; for example, the function $\exp(1/z)$ never takes on the value ⁠ $0$ ⁠.)

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Sources and credits

This article is adapted from the Wikipedia article “Essential singularity”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Essential singularity.png by Functor Salad, CC BY-SA 3.0

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