Calculus How To

Punctured Disk: Simple Definition

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Imagine a flat disk the size of the milky way, then take a (very thin) pin and prick through the disk in the exact center; You’ve just created a punctured disk (sometimes called a deleted disk).

punctured disk 2

Punctured open disk around z0(left); Open disk around z0 (right).

More technically, this type of disk is an annulus (a flat donut) with an inner radius of zero, i.e. a disk without a center. However, there is more than one definition. For example Nevanlinna (1970) regarded the point of puncture as when a boundary curve shrinks to a single point. This makes sense when you think of a black hole, which is a gravitational singularity with an enormous mass in an infinitely small space. Disks can also be punctured more than once (Beardon, 1983), so you may see them described as once-punctured or twice punctured.

Punctured disks arise in complex analysis when studying functions that are well-behaved except for a central point, where there may be an isolated singularity.

Describing the a Punctured Disk

A punctured disk can be described as an open annulus with center x0 and an inner radius of zero. The region is defined by two inequalities

0 < |z – z0| < R,

where R = the punctured disk’s radius.

The radius can be any positive number; when R is infinity, the region is usually called a punctured plane (Sarason, 2007).

A few interesting features of punctured disks:


  • Although punctured disks are sometimes described as an open annulus, the disks themselves are neither closed sets nor open sets,
  • The disks aren’t simply connected. The puncture prevents a closed curve around the region from contracting to a point while keeping within the region (MIT).
  • A punctured disk can also be written in terms of a Laurent series.

References

Beardon, A. (1983). The geometry of discrete groups, Graduate Texts in Mathematics 91, Springer-Verlag, Berlin.
MIT OpenCourseware. (2010). 18.02SC Multivariable Calculus. Retrieved September 27, 2020 from: https://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/3.-double-integrals-and-line-integrals-in-the-plane/part-c-greens-theorem/session-72-simply-connected-regions-and-conservative-fields/MIT18_02SC_pb_72_comb.pdf
Nevanlinna, R. (1970). Analytic functions, Springer-Verlag, Berlin.
Singularities. Article Posted on website The Physics of The Universe.
Sarason, D. (2007). Complex Function Theory. American Mathematical Society.


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