Math 2.0

Functions of a complex variable Complex integration Theorem

Uniform limits of holomorphic functions are holomorphic

Builds on Holomorphic functionMorera's theoremCauchy's integral theorem

Statement

Let U⊆ℂU \subseteq \mathbb{C} be an open set and let f1,f2,…f_1, f_2, \dots be functions U→ℂU \to \mathbb{C} that are holomorphic on UU. Let f:U→ℂf : U \to \mathbb{C} be a function such that fn→ff_n \to f uniformly on each compact subset of UU. Then ff is holomorphic on UU. Moreover, fn′→f′f_n' \to f' uniformly on each compact subset of UU.

Remarks

Proof idea: ff is continuous. On each disc in UU, Cauchy's integral theorem gives ∮∂Tfn=0\oint_{\partial T} f_n = 0 for each triangle TT, so ∮∂Tf=0\oint_{\partial T} f = 0 by the uniform convergence, and Morera's theorem applies. A main use: if a series ∑gk\sum g_k of holomorphic functions converges uniformly on compact subsets of UU, then its sum is holomorphic on UU. There is no real analogue. On [−1,1][-1, 1], the polynomials converge uniformly to ∣x∣|x|, which is not differentiable at 00.

Reading path

To understand this page from the ground up, read these 19 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 abovedefinitionContour integral, definition, no evidence levelContour integraldefinitionReal 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 functiondefinitionGoursat's theorem, statement, no evidence levelGoursat's theoremstatementCauchy's integral theorem, statement, no evidence levelCauchy's integral the…statementHolomorphic functions on a disc have primitives, statement, no evidence levelHolomorphic functions…statementMorera's theorem, statement, no evidence levelMorera's theoremstatementUniform limits of holomorphic functions are holomorphic, statement, no evidence levelUniform limits of hol…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.
  • no evidence level
  1. Complex numbers object
  2. Complex plane concept
  3. Real numbers object
  4. Set bounded above definition
  5. Contour integral definition
  6. Real and imaginary parts definition
  7. Supremum definition
  8. Completeness axiom statement
  9. Existence of square roots of nonnegative reals statement
  10. Square root definition
  11. Modulus of a complex number definition
  12. Complex differentiability definition
  13. Domain in the complex plane definition
  14. Holomorphic function definition
  15. Goursat's theorem statement
  16. Cauchy's integral theorem statement
  17. Holomorphic functions on a disc have primitives statement
  18. Morera's theorem statement
  19. Uniform limits of holomorphic functions are holomorphic statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Morera%27s_theorem, Uniform limits
  2. Web https://en.wikipedia.org/wiki/Holomorphic_function, Properties
About this page
Tag0431RN0R1N
URIm2://0431RN0R1N
Aliasesweierstrass-convergence-theorem
Kindstatement, theorem
WritingW0 stub
MSC30A99
StatusThis copy of the site has no build data. Run m2 status 0431RN0R1N.
UsesHolomorphic function, Morera's theorem, Cauchy's integral theorem

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