Math 2.0

Functions of a complex variable Complex integration Theorem

Cauchy's integral theorem

Builds on Holomorphic functionContour integralDomain in the complex plane

Statement

Let U⊆ℂU \subseteq \mathbb{C} be a simply connected open set. This means that UU is connected, and each closed path in UU can be continuously deformed in UU to a constant path. Let f:U→ℂf : U \to \mathbb{C} be holomorphic on UU. Let γ:[a,b]→U\gamma : [a, b] \to U be a closed path that is continuous and piecewise continuously differentiable. Then the contour integral of ff along γ\gamma is ∮γf(z) dz=0.\oint_\gamma f(z) \, dz = 0.

Remarks

Open discs, open half-planes, ℂ\mathbb{C} and each convex open set are simply connected. So the theorem holds on each of them. A special case: if ff is holomorphic on an open set that contains the closed disc ∣z−c∣≤R|z - c| \le R, then ∮∣z−c∣=Rf(z) dz=0\oint_{|z - c| = R} f(z) \, dz = 0. The hypothesis on UU is necessary. On ℂ∖{0}\mathbb{C} \setminus \{0\}, the integral of 1/z over the unit circle is 2πi2 \pi i. It follows that the integral of ff along a path in UU depends only on the end points of the path.

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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 functiondefinitionCauchy's integral theorem, statement, no evidence levelCauchy's integral the…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.
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  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. Cauchy's integral theorem statement

Where this leads

3 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Cauchy%27s_integral_theorem, Formulation on simply connected regions
  2. Web https://en.wikipedia.org/wiki/Cauchy%27s_integral_theorem, Statement
About this page
Tag01MPZXYVYF
URIm2://01MPZXYVYF
Aliasescauchy-goursat-theorem, cauchy-integral-theorem
Kindstatement, theorem
WritingW1 stated
MSC30E20
StatusThis copy of the site has no build data. Run m2 status 01MPZXYVYF.
UsesHolomorphic function, Contour integral, Domain in the complex plane
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