Essential singularity

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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).
Model illustrating essential singularity of a complex function 6w=exp(1/(6z))

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[edit]

Consider an open subset of the complex plane . Let be an element of , and a holomorphic function. The point is called an essential singularity of the function if the singularity is neither a pole nor a removable singularity.

For example, the function has an essential singularity at .

Alternate descriptions[edit]

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     exists, 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). A related definition is that if there is a point for which no derivative of converges to a limit as tends to , then is an essential singularity of .[1]

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.)

References[edit]

  1. ^ Weisstein, Eric W. "Essential Singularity". MathWorld, Wolfram. Retrieved 11 February 2014.
  • Lars V. Ahlfors; Complex Analysis, McGraw-Hill, 1979
  • Rajendra Kumar Jain, S. R. K. Iyengar; Advanced Engineering Mathematics. Page 920. Alpha Science International, Limited, 2004. ISBN 1-84265-185-4

External links[edit]