Math 2.0

Potential theory Definition

Harmonic conjugate

Builds on Holomorphic functionThe real part of a holomorphic function is harmonic

Definition

Let U⊆ℂU \subseteq \mathbb{C} be an open set, and identify ℂ\mathbb{C} with ℝ2\mathbb{R}^2 by x+iy↔(x,y)x + iy \leftrightarrow (x, y). Let u:U→ℝu : U \to \mathbb{R} be a harmonic function: uu is twice continuously differentiable and uxx+uyy=0u_{xx} + u_{yy} = 0 on UU. A harmonic conjugate of uu is a function v:U→ℝv : U \to \mathbb{R} such that the function f=u+ivf = u + iv is holomorphic on UU.

Remarks

For example, v=2xyv = 2xy is a harmonic conjugate of u=x2−y2u = x^2 - y^2 on ℂ\mathbb{C}, because u+iv=z2u + iv = z^2. If UU is a domain, two harmonic conjugates of uu differ by a real constant. If vv is a harmonic conjugate of uu, then −u-u is a harmonic conjugate of vv, because v−iu=−i(u+iv)v - iu = -i(u + iv) is holomorphic. On a disc, each harmonic function has a harmonic conjugate. On ℂ∖{0}\mathbb{C} \setminus \{0\}, the harmonic function log⁡∣z∣\log |z| has none.

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To understand this page from the ground up, read these 23 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 abovedefinitionComplex exponential, definition, no evidence levelComplex exponentialdefinitionContour integral, definition, no evidence levelContour integraldefinitionPower series in the complex plane, definition, no evidence levelPower series in the c…definitionReal and imaginary parts, definition, no evidence levelReal and imaginary pa…definitionSupremum, definition, no evidence levelSupremumdefinitionCircle integral, definition, no evidence levelCircle integraldefinitionDisc of convergence, definition, no evidence levelDisc of convergencedefinitionCompleteness 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 functiondefinitionCauchy–Riemann equations, statement, no evidence levelCauchy–Riemann equati…statementCauchy's integral formula, statement, no evidence levelCauchy's integral for…statementPower series are holomorphic in the disc of convergence, statement, no evidence levelPower series are holo…statementHolomorphic functions are analytic, statement, no evidence levelHolomorphic functions…statementHolomorphic functions are infinitely differentiable, statement, no evidence levelHolomorphic functions…statementThe real part of a holomorphic function is harmonic, statement, no evidence levelThe real part of a ho…statementHarmonic conjugate, definition, no evidence levelHarmonic conjugatedefinition
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. Complex exponential definition
  5. Contour integral definition
  6. Power series in the complex plane definition
  7. Real and imaginary parts definition
  8. Supremum definition
  9. Circle integral definition
  10. Disc of convergence definition
  11. Completeness axiom statement
  12. Existence of square roots of nonnegative reals statement
  13. Square root definition
  14. Modulus of a complex number definition
  15. Complex differentiability definition
  16. Holomorphic function definition
  17. Cauchy–Riemann equations statement
  18. Cauchy's integral formula statement
  19. Power series are holomorphic in the disc of convergence statement
  20. Holomorphic functions are analytic statement
  21. Holomorphic functions are infinitely differentiable statement
  22. The real part of a holomorphic function is harmonic statement
  23. Harmonic conjugate definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Harmonic_conjugate, lead section
  2. Web https://en.wikipedia.org/wiki/Harmonic_conjugate, Examples
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Kinddefinition
WritingW0 stub
MSC31A05
DefinesHarmonic conjugate
UsesHolomorphic function, The real part of a holomorphic function is harmonic

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