Math 2.0

Functions of a complex variable Counterexample

The complex conjugate is not holomorphic

Builds on Complex conjugateComplex differentiabilityHolomorphic function

Counterexample

Let f:ℂ→ℂf : \mathbb{C} \to \mathbb{C} be the complex conjugate map, f(z)=zˉf(z) = \bar z. The function ff is continuous, but it is not complex differentiable at any point z0∈ℂz_0 \in \mathbb{C}. Thus ff is not holomorphic on any nonempty open set. Proof: the difference quotient at z0z_0 is f(z0+h)−f(z0)h=hˉh.\frac{f(z_0 + h) - f(z_0)}{h} = \frac{\bar h}{h}. It is 11 for each real h≠0h \neq 0 and −1-1 for each purely imaginary h≠0h \neq 0. So it has no limit as h→0h \to 0.

Remarks

The real and imaginary parts u=xu = x and v=−yv = -y are differentiable, but ux=1u_x = 1 and vy=−1v_y = -1. So the Cauchy–Riemann equations fail at each point. As a map of ℝ2\mathbb{R}^2, ff is the reflection in the real axis. It is real linear but not complex linear. For the same reason, Re⁡z\operatorname{Re} z, Im⁡z\operatorname{Im} z and ∣z∣|z| are not holomorphic on any nonempty open set.

Reading path

To understand this page from the ground up, read these 13 pages in order. The foundations come first.

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Complex numbers, object, no evidence levelComplex numbersobjectReal numbers, object, no evidence levelReal numbersobjectSet bounded above, definition, no evidence levelSet bounded abovedefinitionReal and imaginary parts, definition, no evidence levelReal and imaginary pa…definitionSupremum, definition, no evidence levelSupremumdefinitionComplex conjugate, definition, no evidence levelComplex conjugatedefinitionCompleteness 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 functiondefinitionThe complex conjugate is not holomorphic, counterexample, no evidence levelThe complex conjugate…counterexample
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.
  • no evidence level
  1. Complex numbers object
  2. Real numbers object
  3. Set bounded above definition
  4. Real and imaginary parts definition
  5. Supremum definition
  6. Complex conjugate 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. The complex conjugate is not holomorphic counterexample

Further reading

  1. Web https://en.wikipedia.org/wiki/Holomorphic_function, Examples
  2. Web https://en.wikipedia.org/wiki/Cauchy%E2%80%93Riemann_equations, Independence of the complex conjugate
About this page
Tag0Z8ATBPM5W
URIm2://0Z8ATBPM5W
Aliasesz-conjugate-is-not-holomorphic
Kindcounterexample
WritingW0 stub
MSC30A99
UsesComplex conjugate, Complex differentiability, Holomorphic function

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