Incomplex analysis, anessential singularity of afunction is a "severe"singularity near which the function exhibits striking behavior.

The categoryessential singularity is a "left-over" or default group ofisolated 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 andpoles. In practice some^[who?] include non-isolated singularities too; those do not have aresidue.

Consider anopen subset${\displaystyle U}$ of thecomplex plane⁠${\displaystyle \mathbb{C}}$⁠. Let${\displaystyle a}$ be an element of⁠${\displaystyle U}$⁠, and${\displaystyle f:U\setminus \{a\}\to \mathbb{C}}$ aholomorphic function. The point${\displaystyle a}$ is called anessential singularity of the function${\displaystyle f}$ if the singularity is neither apole nor aremovable singularity.

For example, the function${\displaystyle f(z)=e^{1/z}}$ has an essential singularity at⁠${\displaystyle z=0}$⁠.

Let${\displaystyle a}$ be acomplex number, and assume that${\displaystyle f(z)}$ is not defined at${\displaystyle a}$ but isanalytic in some region${\displaystyle U}$ of the complex plane, and that everyopenneighbourhood of${\displaystyle a}$ has non-empty intersection with⁠${\displaystyle U}$⁠.

• If both${\displaystyle \underset{z\to a}{lim}f(z)}$ and${\displaystyle \underset{z\to a}{lim}1/f(z)}$ exist, then${\displaystyle a}$ is aremovable singularity of both${\displaystyle f}$ and⁠${\displaystyle 1/f}$⁠.
• If${\displaystyle \underset{z\to a}{lim}f(z)}$ exists but${\displaystyle \underset{z\to a}{lim}1/f(z)}$ does not exist (⁠${\displaystyle \underset{z\to a}{lim}\left|1/f(z)\right|=\mathrm{\infty }}$⁠), then${\displaystyle a}$ is azero of${\displaystyle f}$ and apole of⁠${\displaystyle 1/f}$⁠.
• If${\displaystyle \underset{z\to a}{lim}f(z)}$ does not exist (in fact${\displaystyle \underset{z\to a}{lim}|f(z)|=\mathrm{\infty }}$) but${\displaystyle \underset{z\to a}{lim}1/f(z)}$ exists, then${\displaystyle a}$ is apole of${\displaystyle f}$ and azero of⁠${\displaystyle 1/f}$⁠.
• If neither${\displaystyle \underset{z\to a}{lim}f(z)}$ nor${\displaystyle \underset{z\to a}{lim}1/f(z)}$ exists, then${\displaystyle a}$ is anessential singularity of both${\displaystyle f}$ and⁠${\displaystyle 1/f}$⁠.

Another way to characterize an essential singularity is that theLaurent series of${\displaystyle f}$ at the point${\displaystyle a}$ has infinitely many negative degree terms (i.e., theprincipal part of the Laurent series is an infinite sum). A related definition is that if there is a point${\displaystyle a}$ for which${\displaystyle f(z)(z-a{)}^n}$ is not differentiable for any integer⁠${\displaystyle n>0}$⁠, then${\displaystyle a}$ is an essential singularity of⁠${\displaystyle f}$⁠.^[1]

On aRiemann sphere with apoint at infinity,⁠${\displaystyle {\mathrm{\infty }}_{\mathbb{C}}}$⁠, the function${\displaystyle f(z)}$ has an essential singularity at that point if and only if the${\displaystyle f(1/z)}$ has an essential singularity at⁠${\displaystyle 0}$⁠: i.e. neither${\displaystyle \underset{z\to 0}{lim}f(1/z)}$ nor${\displaystyle \underset{z\to 0}{lim}1/f(1/z)}$ exists.^[2] TheRiemann zeta function on the Riemann sphere has only one essential singularity, which is at${\displaystyle {\mathrm{\infty }}_{\mathbb{C}}}$.^[3] Indeed, everymeromorphic function aside that is not arational function has a unique essential singularity at⁠${\displaystyle {\mathrm{\infty }}_{\mathbb{C}}}$⁠.

The behavior ofholomorphic functions near their essential singularities is described by theCasorati–Weierstrass theorem and by the considerably strongerPicard's great theorem. The latter says that in every neighborhood of an essential singularity⁠${\displaystyle a}$⁠, the function${\displaystyle f}$ takes onevery complex value, except possibly one, infinitely many times. (The exception is necessary; for example, the function${\displaystyle \exp (1/z)}$ never takes on the value⁠${\displaystyle 0}$⁠.)

• An Essential Singularity byStephen Wolfram,Wolfram Demonstrations Project.
• Essential Singularity on Planet Math

• Complex analysis

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