Math 2.0

Functions of a complex variable Complex integration Theorem

Morera's theorem

Builds on Holomorphic functionContour integralHolomorphic functions on a disc have primitives

Statement

Let U⊆ℂU \subseteq \mathbb{C} be an open set and let f:U→ℂf : U \to \mathbb{C} be continuous. Assume that ∮∂Tf(z) dz=0\oint_{\partial T} f(z) \, dz = 0 for each closed triangle T⊆UT \subseteq U, where ∂T\partial T is the boundary of TT traveled once and the integral is a contour integral. Then ff is holomorphic on UU.

Remarks

It is enough to know that ∮γf(z) dz=0\oint_\gamma f(z) \, dz = 0 for each closed piecewise continuously differentiable path γ\gamma in UU, because the boundary of a triangle is such a path. Proof idea: on each disc in UU, F(z)=∫[c,z]f(w) dwF(z) = \int_{[c, z]} f(w) \, dw is a primitive of ff. So FF is holomorphic, and f=F′f = F' is holomorphic because derivatives of holomorphic functions are holomorphic. Together with Goursat's theorem, this shows: a continuous ff is holomorphic on UU if and only if its integrals over all triangles in UU are zero. A main use is the theorem on uniform limits of holomorphic functions.

Reading path

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Complex numbers, object, no evidence levelComplex numbersobjectReal 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…definitionHolomorphic function, definition, no evidence levelHolomorphic functiondefinitionGoursat's theorem, statement, no evidence levelGoursat's theoremstatementHolomorphic functions on a disc have primitives, statement, no evidence levelHolomorphic functions…statementMorera's theorem, statement, no evidence levelMorera's 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.
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  1. Complex numbers object
  2. Real numbers object
  3. Set bounded above definition
  4. Contour integral definition
  5. Real and imaginary parts definition
  6. Supremum definition
  7. Completeness axiom statement
  8. Existence of square roots of nonnegative reals statement
  9. Square root definition
  10. Modulus of a complex number definition
  11. Complex differentiability definition
  12. Holomorphic function definition
  13. Goursat's theorem statement
  14. Holomorphic functions on a disc have primitives statement
  15. Morera's theorem statement

Where this leads

3 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Morera%27s_theorem, lead section
  2. Web https://en.wikipedia.org/wiki/Morera%27s_theorem, Weakening of hypotheses
About this page
Tag0VDD7W0EKZ
URIm2://0VDD7W0EKZ
Aliasesmoreras-theorem
Kindstatement, theorem
WritingW1 stated
MSC30E20
StatusThis copy of the site has no build data. Run m2 status 0VDD7W0EKZ.
UsesHolomorphic function, Contour integral, Holomorphic functions on a disc have primitives
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