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 striking behavior.

File:Essential singularity.png

Plot of the function exp(1/z), centered on the essential singularity at z = 0. The hue represents the complex argument, the luminance represents the absolute value. This plot shows how approaching the essential singularity from different directions yields different behaviors (as opposed to a pole, which, approached from any direction, would be uniformly white).

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 someTemplate:Who? include non-isolated singularities too; those do not have a residue.

Formal description

Consider an open subset $U$ of the complex plane Template:Tmath. Let $a$ be an element of Template:Tmath, and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(z)=e^{1/z}} has an essential singularity at Template:Tmath.

Alternative descriptions

Let $a$ be a complex number, and assume that Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 Template:Tmath.

• If both Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \lim _{z\to a}f(z)} and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \lim _{z\to a}{1}/{f(z)}} exist, then $a$ is a removable singularity of both $f$ and Template:Tmath.
• If Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \lim _{z\to a}f(z)} exists but Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \lim _{z\to a}{1}/{f(z)}} does not exist (Template:Tmath), then $a$ is a zero of $f$ and a pole of Template:Tmath.
• If $\lim _{z\to a}f(z)$ does not exist (in fact Template:Tmath) but Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \lim _{z\to a}{1}/{f(z)}} exists, then $a$ is a pole of $f$ and a zero of Template:Tmath.
• If neither $\lim _{z\to a}f(z)$ nor Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \lim _{z\to a}{1}/{f(z)}} exists, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} is an essential singularity of both Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} and Template:Tmath.

Another way to characterize an essential singularity is that the Laurent series of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} at the point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} for which Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(z)(z-a)^n} is not differentiable for any integer Template:Tmath, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} is an essential singularity of Template:Tmath.

On a Riemann sphere with a point at infinity, Template:Tmath, the function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {f(z)}} has an essential singularity at that point if and only if the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {f(1/z)}} has an essential singularity at Template:Tmath: i.e. neither Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{z \to 0}{f(1/z)}} nor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Template:Tmath. Indeed, every meromorphic function aside that is not a rational function has a unique essential singularity at Template:Tmath.

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 Template:Tmath, the function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} takes on every complex value, except possibly one, infinitely many times. (The exception is necessary; for example, the function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \exp(1/z)} never takes on the value Template:Tmath.)

References

• Ahlfors, Lars V. (1979), Complex Analysis, McGraw-Hill
• Jain, Rajendra Kumar; Iyengar, S. R. K. (2004), Advanced Engineering Mathematics, Alpha Science International, Limited, p. 920, ISBN 1-84265-185-4

External links

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