# Essential Singularity

Also found in: Wikipedia.

## essential singularity

[i′sen·chəl siŋ·gyə′lar·əd·ē]
(mathematics)
An isolated singularity of a complex function which is neither removable nor a pole.

## Essential Singularity

If a function is single-valued and analytic in some neighborhood of the point z0 in the complex plane and there exists neither a finite nor an infinite limit for the function as zz0, then z0 is said to be an essential singularity of the function (seeANALYTIC FUNCTIONS). For example, the point z = 0 is an essential singularity of such function as e1/z, z sin (1/z), and cos (1/z) + 1n (z + 1).

In a neighborhood of an essential singularity z0, the function f(z) can be expanded in a Laurent series:

Here, infinitely many of the numbers b1, b2, ... are nonzero. This property is often used to identify essential singularities.

The behavior of a function in the neighborhood of an essential singularity can be dealt with on the basis of the Casorati-Weier-strass theorem. A generalization of this theorem is provided by Picard’s big, or second, theorem: in every neighborhood of an essential singularity of an analytic function the function takes on every value, with at most one exception. Picard’s theorem has a number of extensions and refinements.

In some branches of the theory of analytic functions, the term “essential singularity” is also applied to singularities of a more complex nature.

### REFERENCES

Markushevich, A. I. Teoriia analiticheskikh funktsii, 2nd ed., vols. 1–2. Moscow, 1967–68.
Nevanlinna, R. Odnoznachnye analiticheskie funktsii. Moscow-Leningrad, 1941. (Translated from German.)
References in periodicals archive ?
gives the appearance of approaching some essential singularity [italicized for emphasis] in the history of the race beyond which human affairs, as we know them, could not continue" (Ulam, 1958).
In all our calculation we have seen only one singularity which is the essential singularity and occur at (eq.) These spacetimes will help in understanding of the gravitational wave spacetime, black hole [3] and asymptotic behavior of black hole [4].
It is evident from the graphs of the functions (eqs.) given in the following figures, that there is always a singularity at (eq.) in all cases, which is the essential singularity, and there is no other singularity in these cases [6].
In 1958, the US mathematician, physicist, inventor and polymath John Von Neumann described how technological changes and the ever-accelerating progress of technology give the appearance of our approaching some essential singularity beyond which human affairs as we know them could not continue.
In a 1958 interview, von Neumann described the "ever accelerating progress of technology and changes in the mode of human life, which gives the appearance of approaching some essential singularity in the history of the race beyond which human affairs, as we know them, cannot continue."
Consider the simplest example of an essential singularity on [T.sub.[rho]]:
Observe that in the case of an essential singularity of S the asymptotic behavior of the Cauchy transform
Private betrothals, nocturnal weddings, and consummations of unions prior to official church nuptials were roundly condemned by the provincial Catholic council meeting in Reims in 1583, by the 1587 synodal statutes for the diocese of Orleans, and by the new ritual book for the diocese of Saint Brieu redacted in 1606.(72) Cohabitations of any kind by couples prior to their church wedding before a priest were reclassified as sins meriting excommunication in the 1606 edition of the ritual book authorized by the bishop of Evreux and in the Rituels de Paris for 1615 and 1630.(73) Multiple church services joining the same couple in matrimony were prohibited and the essential singularity of the commitment articulated before a priest stressed.

Site: Follow: Share:
Open / Close