Math 2.0

Functions of a complex variable Definition

Domain in the complex plane

Builds on Complex planeModulus of a complex number

Definition

Let DD be a subset of the complex plane ℂ\mathbb{C}. The set DD is open if for each z0∈Dz_0 \in D there is a real number r>0r > 0 such that each z∈ℂz \in \mathbb{C} with ∣z−z0∣<r|z - z_0| < r is in DD. Here ∣z−z0∣|z - z_0| is the modulus of z−z0z - z_0. The set DD is connected if there are no two disjoint nonempty open sets AA and BB with D=A∪BD = A \cup B. A domain in the complex plane is a subset D⊆ℂD \subseteq \mathbb{C} that is nonempty, open and connected.

Remarks

An open set D⊆ℂD \subseteq \mathbb{C} is connected if and only if each two points of DD are the end points of a polygonal path in DD. Examples of domains are ℂ\mathbb{C}, each open disc, the upper half-plane {z:Im⁡z>0}\{z : \operatorname{Im} z > 0\} and ℂ∖{0}\mathbb{C} \setminus \{0\}. The union of two disjoint open discs is open but not connected, so it is not a domain. Some authors use the word region for a domain. Other authors use region for a set between a domain and its closure.

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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 …definitionDomain in the complex plane, definition, no evidence levelDomain in the complex…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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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. Domain in the complex plane definition

Where this leads

43 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Domain_(mathematical_analysis), lead section
  2. Web https://en.wikipedia.org/wiki/Domain_(mathematical_analysis), Conventions
About this page
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Kinddefinition
WritingW0 stub
MSC30A99
DefinesDomain in the complex plane
UsesComplex plane, Modulus of a complex number

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