Math 2.0

Functions of a complex variable Singularities and residues Theorem

Casorati–Weierstrass theorem

Builds on Essential singularityRiemann's theorem on removable singularities

Statement

Let c∈ℂc \in \mathbb{C}, let r>0r > 0, and let ff be holomorphic on the punctured disc 0<∣z−c∣<r0 < |z - c| < r. Assume that cc is an essential singularity of ff. Then for each δ\delta with 0<δ<r0 < \delta < r, the image f({z:0<∣z−c∣<δ})f(\{z : 0 < |z - c| < \delta\}) is dense in ℂ\mathbb{C}. Thus for each w∈ℂw \in \mathbb{C} there is a sequence zn→cz_n \to c with zn≠cz_n \neq c and f(zn)→wf(z_n) \to w.

Remarks

Proof idea: if the image of a punctured disc misses a disc ∣w−w0∣<ε|w - w_0| < \varepsilon, then g(z)=1/(f(z)−w0)g(z) = 1/(f(z) - w_0) is bounded near cc. By Riemann's theorem, gg is holomorphic at cc, and then cc is removable or a pole of ff. Picard's great theorem is a much stronger statement. In the Russian literature, the theorem has the name of Sokhotski.

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Complex numbers, object, no evidence levelComplex numbersobjectReal 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…definitionHolomorphic function, definition, no evidence levelHolomorphic functiondefinitionIsolated singularity, definition, no evidence levelIsolated singularitydefinitionPole, definition, no evidence levelPoledefinitionRemovable singularity, definition, no evidence levelRemovable singularitydefinitionEssential singularity, definition, no evidence levelEssential singularitydefinitionRiemann's theorem on removable singularities, statement, no evidence levelRiemann's theorem on …statementCasorati–Weierstrass theorem, statement, no evidence levelCasorati–Weierstrass …statement
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. Real numbers object
  3. Set bounded above definition
  4. Real and imaginary parts definition
  5. Supremum definition
  6. Completeness axiom statement
  7. Existence of square roots of nonnegative reals statement
  8. Square root definition
  9. Modulus of a complex number definition
  10. Complex differentiability definition
  11. Holomorphic function definition
  12. Isolated singularity definition
  13. Pole definition
  14. Removable singularity definition
  15. Essential singularity definition
  16. Riemann's theorem on removable singularities statement
  17. Casorati–Weierstrass theorem statement

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Further reading

  1. Web https://en.wikipedia.org/wiki/Casorati%E2%80%93Weierstrass_theorem, Formal statement of the theorem
  2. Web https://en.wikipedia.org/wiki/Casorati%E2%80%93Weierstrass_theorem, Proof of the theorem
About this page
Tag02HHNPC8ED
URIm2://02HHNPC8ED
Aliasessokhotski-theorem, casorati-weierstrass-sokhotski-theorem
Kindstatement, theorem
WritingW1 stated
MSC30D30
StatusThis copy of the site has no build data. Run m2 status 02HHNPC8ED.
UsesEssential singularity, Riemann's theorem on removable singularities
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