Math 2.0

Functions of a complex variable Complex integration Theorem

Open mapping theorem

Builds on Holomorphic functionDomain in the complex plane

Statement

Let D⊆ℂD \subseteq \mathbb{C} be a domain and let f:D→ℂf : D \to \mathbb{C} be holomorphic on DD. Assume that ff is not constant on DD. Then ff is an open map: for each open set V⊆DV \subseteq D, the image f(V)f(V) is an open subset of ℂ\mathbb{C}.

Remarks

The maximum modulus principle follows. If ∣f∣|f| had a local maximum at z0z_0, then ff would map a small disc around z0z_0 onto a set that contains no disc around f(z0)f(z_0). The connectedness of DD matters only to exclude functions that are constant on one part of DD. There is no real analogue: x↦x2x \mapsto x^2 maps (−1,1)(-1, 1) onto [0,1)[0, 1), which is not open in ℝ\mathbb{R}. In contrast, z↦z2z \mapsto z^2 maps the unit disc onto the unit disc.

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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 …definitionComplex differentiability, definition, no evidence levelComplex differentiabi…definitionDomain in the complex plane, definition, no evidence levelDomain in the complex…definitionHolomorphic function, definition, no evidence levelHolomorphic functiondefinitionOpen mapping theorem, statement, no evidence levelOpen mapping theoremstatement
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. Complex differentiability definition
  12. Domain in the complex plane definition
  13. Holomorphic function definition
  14. Open mapping theorem statement

Where this leads

6 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Open_mapping_theorem_(complex_analysis), lead section
  2. Web https://en.wikipedia.org/wiki/Open_mapping_theorem_(complex_analysis), Proof
About this page
Tag03YEM7JEDN
URIm2://03YEM7JEDN
Aliasesopen-mapping-theorem-complex-analysis
Kindstatement, theorem
WritingW1 stated
MSC30A99
StatusThis copy of the site has no build data. Run m2 status 03YEM7JEDN.
UsesHolomorphic function, Domain in the complex plane
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