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Isolated Singularity

An isolated singularity is a singularity for which there exists a positive real number epsilon such that there are no other singularities within a neighborhood of radius epsilon centered about the singularity.

In complex analysis, a is an isolated singularity of a function f when f is analytic on some deleted neighborhood 0<|z-a|<R. The Laurent series there classifies a as a removable singularity, a pole, or an essential singularity (Gautam 2020, pp. 3-4). All three cases have a well-defined complex residue, namely the coefficient of (z-a)^(-1).

The types of isolated singularities possible for cubic surfaces have been classified (Schläfli 1863, Cayley 1869, Bruce and Wall 1979) and are summarized in the following table from Fischer (1986).

namesymbolnormal formCoxeter-Dynkin diagram
conic double pointC_2x^2+y^2+z^2A_1
biplanar double pointB_3x^2+y^2+z^3A_2
biplanar double pointB_4x^2+y^2+z^4A_3
biplanar double pointB_5x^2+y^2+z^5A_4
biplanar double pointB_6x^2+y^2+z^6A_5
uniplanar double pointU_6x^2+z(y^2+z^2)D_4
uniplanar double pointU_7x^2+z(y^2+z^3)D_5
uniplanar double pointU_8x^2+y^3+z^4E_6
elliptic cone point--xy^2-4z^3-g_2x^2y+g_3x^3E^~_6

See also

Complex Residue, Cubic Surface, Essential Singularity, Laurent Series, Pole, Rational Double Point, Removable Singularity, Singular Point

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References

Bruce, J. and Wall, C. T. C. "On the Classification of Cubic Surfaces." J. London Math. Soc. 19, 245-256, 1979.Cayley, A. "A Memoir on Cubic Surfaces." Phil. Trans. Roy. Soc. 159, 231-326, 1869.Fischer, G.(Ed.). Mathematische Modelle aus den Sammlungen von Universitäten und Museen, Kommentarband. Braunschweig, Germany: Vieweg, pp. 12-13, 1986.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.Krantz, S. G. Handbook of Complex Variables. Boston, MA: Birkhäuser, p. 41, 1999.Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 380-381, 1953.Schläfli, L. "On the Distribution of Surfaces of Third Order into Species, in Reference to the Absence or Presence of Singular Points, and the Reality of Their Lines." Philos. Trans. Roy. Soc. London 153, 193-241, 1863.

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Isolated Singularity

Cite this as:

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

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