Math 2.0

Functions of a complex variable Complex integration Theorem

Goursat's theorem

Builds on Holomorphic functionContour integral

Statement

Let U⊆ℂU \subseteq \mathbb{C} be an open set and let f:U→ℂf : U \to \mathbb{C} be holomorphic on UU. Let Q={x+iy:a1≤x≤b1, a2≤y≤b2}Q = \{x + iy : a_1 \le x \le b_1, \ a_2 \le y \le b_2\} be a closed rectangle with Q⊆UQ \subseteq U, where a1<b1a_1 < b_1 and a2<b2a_2 < b_2 are real numbers. Let ∂Q\partial Q be the boundary of QQ, traveled once counterclockwise. Then the contour integral of ff along ∂Q\partial Q is ∮∂Qf(z) dz=0.\oint_{\partial Q} f(z) \, dz = 0.

Remarks

The theorem does not assume that f′f' is continuous. This is the difference with the older proof by Green's theorem. The same statement holds for each closed triangle T⊆UT \subseteq U and its boundary. Proof idea: divide QQ into four equal rectangles and keep one where the integral is largest. Repeat, and use the complex derivative at the limit point. Goursat's theorem is the first step to primitives on a disc and to Cauchy's integral theorem.

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 theoremstatement
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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

Where this leads

10 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Cauchy%27s_integral_theorem, Discussion
  2. Web https://en.wikipedia.org/wiki/Cauchy%E2%80%93Riemann_equations, Goursat's theorem and its generalizations
About this page
Tag0BRG85WXYC
URIm2://0BRG85WXYC
Aliasesgoursat-lemma
Kindstatement, theorem
WritingW0 stub
MSC30E20
StatusThis copy of the site has no build data. Run m2 status 0BRG85WXYC.
UsesHolomorphic function, Contour integral

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