Math 2.0

Functions of a complex variable Theorem

Isolated zeros theorem

Builds on Holomorphic functionHolomorphic functions are analyticOrder of a zero

Statement

Let U⊆ℂU \subseteq \mathbb{C} be an open set, let f:U→ℂf : U \to \mathbb{C} be holomorphic on UU, and let z0∈Uz_0 \in U. Then exactly one of these two cases holds.

  1. There is r>0r > 0 such that f(z)=0f(z) = 0 for each zz with ∣z−z0∣<r|z - z_0| < r.
  2. There is r>0r > 0 such that f(z)≠0f(z) \neq 0 for each zz with 0<∣z−z0∣<r0 < |z - z_0| < r.

Remarks

Proof: if case 1 does not hold, then f(z)=(z−z0)mg(z)f(z) = (z - z_0)^m g(z) near z0z_0 with g(z0)≠0g(z_0) \neq 0, where mm is the order of ff at z0z_0. By continuity, g≠0g \neq 0 near z0z_0. On a domain, it follows that the zeros of a holomorphic function that is not identically zero have no accumulation point in the domain. The real function x↦e−1/x2x \mapsto e^{-1/x^2}, with value 00 at 00, is infinitely differentiable and has all derivatives 00 at 00. The theorem has no analogue for such real functions.

Reading path

To understand this page from the ground up, read these 23 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 abovedefinitionComplex exponential, definition, no evidence levelComplex exponentialdefinitionContour integral, definition, no evidence levelContour integraldefinitionPower series in the complex plane, definition, no evidence levelPower series in the c…definitionReal and imaginary parts, definition, no evidence levelReal and imaginary pa…definitionSupremum, definition, no evidence levelSupremumdefinitionCircle integral, definition, no evidence levelCircle integraldefinitionDisc of convergence, definition, no evidence levelDisc of convergencedefinitionCompleteness 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 functiondefinitionCauchy's integral formula, statement, no evidence levelCauchy's integral for…statementZero of a holomorphic function, definition, no evidence levelZero of a holomorphic…definitionPower series are holomorphic in the disc of convergence, statement, no evidence levelPower series are holo…statementHolomorphic functions are analytic, statement, no evidence levelHolomorphic functions…statementHolomorphic functions are infinitely differentiable, statement, no evidence levelHolomorphic functions…statementOrder of a zero, definition, no evidence levelOrder of a zerodefinitionIsolated zeros theorem, statement, no evidence levelIsolated zeros 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. Complex exponential definition
  5. Contour integral definition
  6. Power series in the complex plane definition
  7. Real and imaginary parts definition
  8. Supremum definition
  9. Circle integral definition
  10. Disc of convergence definition
  11. Completeness axiom statement
  12. Existence of square roots of nonnegative reals statement
  13. Square root definition
  14. Modulus of a complex number definition
  15. Complex differentiability definition
  16. Holomorphic function definition
  17. Cauchy's integral formula statement
  18. Zero of a holomorphic function definition
  19. Power series are holomorphic in the disc of convergence statement
  20. Holomorphic functions are analytic statement
  21. Holomorphic functions are infinitely differentiable statement
  22. Order of a zero definition
  23. Isolated zeros theorem statement

Where this leads

24 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Identity_theorem, Lemma
About this page
Tag0K83KEJ43P
URIm2://0K83KEJ43P
Aliasesprinciple-of-isolated-zeros, zeros-are-isolated
Kindstatement, theorem
WritingW0 stub
MSC30C15
StatusThis copy of the site has no build data. Run m2 status 0K83KEJ43P.
UsesHolomorphic function, Holomorphic functions are analytic, Order of a zero

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