Math 2.0

Functions of a complex variable Complex integration Theorem

Identity theorem

Builds on Holomorphic functionDomain in the complex planeIsolated zeros theorem

Statement

Let D⊆ℂD \subseteq \mathbb{C} be a domain and let f,g:D→ℂf, g : D \to \mathbb{C} be holomorphic on DD. Let S={z∈D:f(z)=g(z)}S = \{z \in D : f(z) = g(z)\}. Assume that SS has an accumulation point in DD. This means that there is z0∈Dz_0 \in D such that each disc with center z0z_0 contains a point of SS other than z0z_0. Then f(z)=g(z)f(z) = g(z) for each z∈Dz \in D.

Remarks

Proof idea: apply the isolated zeros theorem to f−gf - g. The set of points near which f−g=0f - g = 0 is open, closed in DD and not empty, so it is DD because DD is connected. For example, if ff is entire and f(1/n)=0f(1/n) = 0 for each n≥1n \ge 1, then f=0f = 0. The connectedness of DD is necessary: on the union of two disjoint discs, a function can be 00 on one disc and 11 on the other. The theorem makes analytic continuation unique.

Reading path

To understand this page from the ground up, read these 26 pages in order. The foundations come first.

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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 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…definitionDomain in the complex plane, definition, no evidence levelDomain in the complex…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 theoremstatementIdentity theorem, statement, no evidence levelIdentity 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. Complex plane concept
  3. Real numbers object
  4. Set bounded above definition
  5. Complex exponential definition
  6. Contour integral definition
  7. Power series in the complex plane definition
  8. Real and imaginary parts definition
  9. Supremum definition
  10. Circle integral definition
  11. Disc of convergence definition
  12. Completeness axiom statement
  13. Existence of square roots of nonnegative reals statement
  14. Square root definition
  15. Modulus of a complex number definition
  16. Complex differentiability definition
  17. Domain in the complex plane definition
  18. Holomorphic function definition
  19. Cauchy's integral formula statement
  20. Zero of a holomorphic function definition
  21. Power series are holomorphic in the disc of convergence statement
  22. Holomorphic functions are analytic statement
  23. Holomorphic functions are infinitely differentiable statement
  24. Order of a zero definition
  25. Isolated zeros theorem statement
  26. Identity theorem statement

Where this leads

23 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Identity_theorem, lead section
  2. Web https://en.wikipedia.org/wiki/Identity_theorem, Proof
About this page
Tag0YBVKMYX0W
URIm2://0YBVKMYX0W
Aliasesuniqueness-theorem-for-holomorphic-functions
Kindstatement, theorem
WritingW1 stated
MSC30C15
StatusThis copy of the site has no build data. Run m2 status 0YBVKMYX0W.
UsesHolomorphic function, Domain in the complex plane, Isolated zeros theorem
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