Essential singularity

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

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))

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 some[who?] include non-isolated singularities too; those do not have a residue.

Formal descriptionEdit

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  .

Alternative descriptionsEdit

Let   be a complex number, assume that   is not defined at   but is analytic in some region   of the complex plane, and that every open neighbourhood of   has non-empty intersection with  .

If both   and   exist, then   is a removable singularity of both   and  .
If   exists but   does not exist (in fact  ), then   is a zero of   and a pole of  .
Similarly, if   does not exist (in fact  ) but   exists, then   is a pole of   and a zero of  .
If neither   nor   exists, then   is an essential singularity of both   and  .

Another way to characterize an essential singularity is that the Laurent series of   at the point   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]

On a Riemann sphere with a point at infinity,  , the function   has an essential singularity at that point if and only if the   has an essential singularity at 0: i.e. neither   nor   exists.[2] The Riemann zeta function on the Riemann sphere has only one essential singularity, at  .[3]

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  , the function   takes on every complex value, except possibly one, infinitely many times. (The exception is necessary; for example, the function   never takes on the value 0.)


  1. ^ Weisstein, Eric W. "Essential Singularity". MathWorld. Wolfram. Retrieved 11 February 2014.
  2. ^ "Infinity as an Isolated Singularity" (PDF). Retrieved 2022-01-06.{{cite web}}: CS1 maint: url-status (link)
  3. ^ Steuding, Jörn; Suriajaya, Ade Irma (2020-11-01). "Value-Distribution of the Riemann Zeta-Function Along Its Julia Lines". Computational Methods and Function Theory. 20 (3): 389–401. doi:10.1007/s40315-020-00316-x. ISSN 2195-3724.
  • 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 linksEdit