Essential Singularity - Alternate Descriptions

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

and exist, then a is a removable singularity of both f and 1/f.

If

exists but does not exist, then a is a zero of f and a pole of 1/f.

Similarly, if

does not exist but does exist, then a is a pole of f and a zero of 1/f.

If neither

nor 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).

The behavior of meromorphic functions near 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.

Read more about this topic:  Essential Singularity

Famous quotes containing the words alternate and/or descriptions:

    Boswell, when he speaks of his Life of Johnson, calls it my magnum opus, but it may more properly be called his opera, for it is truly a composition founded on a true story, in which there is a hero with a number of subordinate characters, and an alternate succession of recitative and airs of various tone and effect, all however in delightful animation.
    James Boswell (1740–1795)

    Matter-of-fact descriptions make the improbable seem real.
    Mason Cooley (b. 1927)