Math 2.0

Functions of a complex variable Theorem

Holomorphic functions on a disc have primitives

Builds on Holomorphic functionGoursat's theoremContour integral

Statement

Let c∈ℂc \in \mathbb{C}, let R∈(0,∞]R \in (0, \infty], and let D={z:∣z−c∣<R}D = \{z : |z - c| < R\} be the open disc with center cc and radius RR. Let f:D→ℂf : D \to \mathbb{C} be holomorphic on DD. Then there is a holomorphic function F:D→ℂF : D \to \mathbb{C} with F′(z)=f(z)F'(z) = f(z) for each z∈Dz \in D.

Remarks

One primitive is F(z)=∫[c,z]f(w) dwF(z) = \int_{[c, z]} f(w) \, dw, the contour integral along the segment from cc to zz. Goursat's theorem for triangles shows that F′=fF' = f. Two primitives of ff on DD differ by a constant, because DD is connected. The disc can be replaced by a convex open set, or more generally by a simply connected open set. The result fails on ℂ∖{0}\mathbb{C} \setminus \{0\}: the function 1/z1/z has no primitive there, by the integral of 1/z.

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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…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. Goursat's theorem statement
  14. Holomorphic functions on a disc have primitives statement

Where this leads

9 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Antiderivative_(complex_analysis), Existence
  2. Web https://en.wikipedia.org/wiki/Antiderivative_(complex_analysis), Uniqueness
About this page
Tag02AC7JR6G6
URIm2://02AC7JR6G6
Aliasesexistence-of-primitives-on-a-disc
Kindstatement, theorem
WritingW0 stub
MSC30E20
StatusThis copy of the site has no build data. Run m2 status 02AC7JR6G6.
UsesHolomorphic function, Goursat's theorem, Contour integral

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