# Essential singularity

Jump to navigation Jump to search 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).

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

## Formal description

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

## 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   $\lim _{z\to a}{\frac {1}{f(z)}}$ exists, then a is a pole of f and a zero of 1/f.

If neither

$\lim _{z\to a}f(z)$ nor   $\lim _{z\to a}{\frac {1}{f(z)}}$ 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). A related definition is that if there is a point $a$ for which no derivative of $f(z)(z-a)^{n}$ converges to a limit as $z$ tends to $a$ , then $a$ is an essential singularity of $f(z)$ .

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, as the function exp(1/z) never takes on the value 0.)