Math 2.0

Functions of a complex variable Definition

Annulus

Builds on Complex planeModulus of a complex number

Definition

Let c∈ℂc \in \mathbb{C} and let r,Rr, R be such that 0≤r<R≤∞0 \le r < R \le \infty. The annulus with center cc, inner radius rr and outer radius RR is the subset A(c;r,R)={z∈ℂ:r<∣z−c∣<R}A(c; r, R) = \{ z \in \mathbb{C} : r < |z - c| < R \} of the complex plane. Here ∣z−c∣|z - c| is the modulus of z−cz - c.

Remarks

Each annulus is a domain. For r=0r = 0, the annulus is the punctured disc 0<∣z−c∣<R0 < |z - c| < R. For r=0r = 0 and R=∞R = \infty, it is ℂ∖{c}\mathbb{C} \setminus \{c\}. For R<∞R < \infty, its area is π(R2−r2)\pi (R^2 - r^2). Holomorphic functions on an annulus have Laurent expansions.

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Complex numbers, object, no evidence levelComplex numbersobjectComplex plane, concept, no evidence levelComplex planeconceptReal numbers, object, no evidence levelReal numbersobjectSet bounded above, definition, no evidence levelSet bounded abovedefinitionReal and imaginary parts, definition, no evidence levelReal and imaginary pa…definitionSupremum, definition, no evidence levelSupremumdefinitionCompleteness axiom, statement, no evidence levelCompleteness axiomstatementExistence of square roots of nonnegative reals, statement, no evidence levelExistence of square r…statementSquare root, definition, no evidence levelSquare rootdefinitionModulus of a complex number, definition, no evidence levelModulus of a complex …definitionAnnulus, definition, no evidence levelAnnulusdefinition
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  1. Complex numbers object
  2. Complex plane concept
  3. Real numbers object
  4. Set bounded above definition
  5. Real and imaginary parts definition
  6. Supremum definition
  7. Completeness axiom statement
  8. Existence of square roots of nonnegative reals statement
  9. Square root definition
  10. Modulus of a complex number definition
  11. Annulus definition

Where this leads

10 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Annulus_(mathematics), Complex structure
About this page
Tag0AJ0YVK743
URIm2://0AJ0YVK743
Kinddefinition
WritingW1 stated
MSC30A99
DefinesAnnulus
UsesComplex plane, Modulus of a complex number

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