Math 2.0

Functions of a complex variable Theorem

Fundamental theorem of contour integration

Builds on Contour integralHolomorphic function

Statement

Let U⊆ℂU \subseteq \mathbb{C} be an open set and let f:U→ℂf : U \to \mathbb{C} be continuous. Let F:U→ℂF : U \to \mathbb{C} be holomorphic with F′(z)=f(z)F'(z) = f(z) for each z∈Uz \in U. Let γ:[a,b]→U\gamma : [a, b] \to U be a continuous and piecewise continuously differentiable path. Then the contour integral of ff along γ\gamma is ∫γf(z) dz=F(γ(b))−F(γ(a)).\int_\gamma f(z) \, dz = F(\gamma(b)) - F(\gamma(a)). In particular, ∫γf(z) dz=0\int_\gamma f(z) \, dz = 0 if γ\gamma is closed.

Remarks

The function FF is a primitive of ff on UU. The integral depends only on the end points of γ\gamma. For each integer n≠−1n \neq -1, znz^n has the primitive zn+1/(n+1)z^{n+1}/(n+1) on ℂ∖{0}\mathbb{C} \setminus \{0\}. So ∮γzn dz=0\oint_\gamma z^n \, dz = 0 for each closed path γ\gamma that does not pass through 00. The function 1/z1/z has no primitive on ℂ∖{0}\mathbb{C} \setminus \{0\}, because its integral over the unit circle is 2πi≠02 \pi i \neq 0.

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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 functiondefinitionFundamental theorem of contour integration, statement, no evidence levelFundamental theorem o…statement
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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. Fundamental theorem of contour integration statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Cauchy%27s_integral_theorem, Fundamental theorem for complex line integrals
  2. Web https://en.wikipedia.org/wiki/Contour_integration, Contour integrals
About this page
Tag0YQ458D7PF
URIm2://0YQ458D7PF
Aliasesfundamental-theorem-for-complex-line-integrals
Kindstatement, theorem
WritingW0 stub
MSC30E20
StatusThis copy of the site has no build data. Run m2 status 0YQ458D7PF.
UsesContour integral, Holomorphic function

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