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Harmonic function

In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function ⁠ f : U → R {\displaystyle f\colon U\to \mathbb {R} } ⁠, where ⁠ U {\displaystyle U} ⁠ is an open subset of ⁠ R n {\displaystyle \mathbb {R} ^{n}} ⁠, that satisfies Laplace's equation, that is, ∂ 2 f ∂ x 1 2 + ∂…

Properties of harmonic functions, Generalizations & Examples

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Etymology of the term "harmonic"

Examples

Properties

Connections with complex function theory

Properties of harmonic functions

Generalizations

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Harmonic function

Nodes77
Edges76
Triples88
Avg. degree1.97
Density0.025974
Components1

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Harmonic function

Top relations

related to External links · 11
Harmonic function → Axler, EMS Press, Encyclopedia, Eric, Harmonic, Harmonic Function Theory, Mathematics, MathWorld, Paul Bourdon, Wade Ramey, Weisstein
related to Connections with complex function theory · 10
Harmonic function → Although, Cauchy, Conversely, Liouville, Omega, Riemann, The, Therefore, They, This
related to Harmonic functions on manifolds · 10
Harmonic function → Beltrami, Delta, Euclidean, Harmonic, Harnack, In, Laplace, Many, Riemannian, With
related to Weakly harmonic function · 10
Harmonic function → Delta, Dirichlet, Dirichlet's, Laplace's, Omega, One, Sobolev, There, This, Weyl's
related to Harmonic forms · 6
Harmonic function → Also, Dirichlet, For, One, Riemannian, This
related to Etymology of the term "harmonic" · 5
Harmonic function → Considering, Fourier, Laplace's, The, These
related to Examples · 5
Harmonic function → Examples, Im, In, The, This
related to Properties · 5
Harmonic function → Delta, If, Laplace, The, The Laplace
related to Harnack's inequality · 4
Harmonic function → Harnack's, Let, Omega, Then
related to Removal of singularities · 4
Harmonic function → If, Omega, Riemann's, The

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Important terminology

harmonic displaystyle function functions mathbb omega property value delta theorem constant two real subset equation principle mean singularities open frac

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
Harmonic functionis atwice continuously differentiable function0.90text
Harmonic functionrelated to Connections with complex function theoryThe0.60section
Harmonic functionrelated to Connections with complex function theoryConversely0.60section
Harmonic functionrelated to Connections with complex function theoryOmega0.60section
Harmonic functionrelated to Connections with complex function theoryThis0.60section
Harmonic functionrelated to Connections with complex function theoryCauchy0.60section
Harmonic functionrelated to Connections with complex function theoryRiemann0.60section
Harmonic functionrelated to Connections with complex function theoryTherefore0.60section
Harmonic functionrelated to Connections with complex function theoryAlthough0.60section
Harmonic functionrelated to Connections with complex function theoryThey0.60section
Harmonic functionrelated to Connections with complex function theoryLiouville0.60section
Harmonic functionrelated to Etymology of the term "harmonic"The0.60section

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