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Essential Singularity

An essential singularity is an isolated singularity of a holomorphic function that is neither a removable singularity nor a pole. If f is analytic on 0<|z-a|<R, then a is an essential singularity iff the Laurent series of f about a has infinitely many nonzero terms with negative powers (Gautam 2020, p. 4). Equivalently, multiplying f(z) by (z-a)^n for any positive integer n does not give a function that extends analytically to a.

For example,

e^(1/z)=sum_(n=0)^infty1/(n!z^n)

has an essential singularity at the origin. It has complex residue 1 there. In contrast, e^(1/z^2) also has an essential singularity at the origin, but its complex residue is 0 since its Laurent series has only even negative powers. A vanishing complex residue therefore does not imply that an isolated singularity is a removable singularity.

See also

Complex Residue, Isolated Singularity, Laurent Series, Picard's Great Theorem, Pole, Removable Singularity, Singularity, Weierstrass-Casorati Theorem

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References

Gautam, S. "Complex Analysis: Lecture 24." Columbus, OH: Ohio State University, pp. 3-5, 2020. https://people.math.osu.edu/gautam.42/S20/LectureNotes/Lecture24.pdf.Knopp, K. "Essential and Non-Essential Singularities or Poles." §31 in Theory of Functions Parts I and II, Two Volumes Bound as One, Part I. New York: Dover, pp. 123-126, 1996.Krantz, S. G. "Removable Singularities, Poles, and Essential Singularities." §4.1.4 in Handbook of Complex Variables. Boston, MA: Birkhäuser, p. 42, 1999.Trott, M. "Residues." §1.6.5 in The Mathematica GuideBook for Symbolics. New York: Springer-Verlag, 2006. https://doi.org/10.1007/0-387-28815-5. https://www.mathematicaguidebooks.org/.

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Essential Singularity

Cite this as:

Weisstein, Eric W. "Essential Singularity." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/EssentialSingularity.html

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