Basic Properties of Harmonic Functions

  • Sheldon Axler
  • Paul Bourdon
  • Wade Ramey
Part of the Graduate Texts in Mathematics book series (GTM, volume 137)


Harmonic functions, for us, live on open subsets of real Euclidean spaces. Throughout this book, n will denote a fixed positive integer greater than 1 and Ω will denote an open, nonempty subset of R n . A twice continuously differentiable, complex-valued function u defined on Ω is harmonic on Ω if
$$\Delta u \equiv 0$$
where Δ = D 1 2 + ⋯ +D n 2 and D j 2 denotes the second partial derivative with respect to the j th coordinate variable. The operator Δ is called the Laplacian, and the equation Δu ≡ 0 is called Laplace’s equation. We say that a function u defined on a (not necessarily open) set ER n is harmonic on E if u can be extended to a function harmonic on an open set containing E.


Power Series Harmonic Function Compact Subset Maximum Principle Dirichlet Problem 
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Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Sheldon Axler
    • 1
  • Paul Bourdon
    • 2
  • Wade Ramey
    • 3
  1. 1.Mathematics DepartmentSan Francisco State UniversitySan FranciscoUSA
  2. 2.Mathematics DepartmentWashington and Lee UniversityLexingtonUSA
  3. 3.BerkeleyUSA

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