Math 2.0

Functions of a complex variable Entire and meromorphic functions Theorem

Picard's great theorem

Builds on Essential singularityCasorati–Weierstrass theorem

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, there is at most one w0∈ℂw_0 \in \mathbb{C} with the following property. The function ff takes the value w0w_0 only finitely many times on 0<∣z−c∣<δ0 < |z - c| < \delta. Thus, on each punctured disc around cc, ff takes each complex value infinitely often, with at most one exception.

Remarks

The exception can occur: e1/ze^{1/z} never takes the value 00, and it takes each w≠0w \neq 0 infinitely often near 00, see exp(1/z). The theorem is much stronger than the Casorati–Weierstrass theorem, which gives only that the values come close to each ww. Picard's little theorem is a related result: a nonconstant entire function takes all complex values with at most one exception.

Reading path

To understand this page from the ground up, read these 18 pages in order. The foundations come first.

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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 …statementPicard's great theorem, statement, no evidence levelPicard's great 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.
  • no evidence level
  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
  18. Picard's great theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Picard_theorem, The theorems
About this page
Tag08QA8QTM71
URIm2://08QA8QTM71
Aliasesbig-picard-theorem, great-picard-theorem
Kindstatement, theorem
WritingW0 stub
MSC30D35
StatusThis copy of the site has no build data. Run m2 status 08QA8QTM71.
UsesEssential singularity, Casorati–Weierstrass theorem

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