KnowraHarmonic functionLinked fromLinked fromThe 17 pages that link to Harmonic function, each with the reason it gives.All 17Broader topic 4Related 9Narrower topic 2Compared with 2Holomorphic functionCompared with: Holomorphic functions yield harmonic components, but harmonicity alone does not ensure holomorphicity.Complex analysisRelated: Real and imaginary parts of holomorphic functions are harmonic, but harmonic functions need not arise globally this way.Laplace's equationBroader topic: These functions are precisely the classical solutions of Laplace's equation.Scalar fieldRelated: Harmonic functions are scalar fields with a distinctive local averaging property.Maximum principleBroader topic: Harmonic functions are the classical solutions for which interior extrema force constancy.Cauchy–Riemann equationsRelated: The equations imply that both components of a sufficiently smooth holomorphic function are harmonic.Potential theoryBroader topic: Harmonic functions are the basic local objects characterized by potential theory.Dirichlet problemBroader topic: For Laplace’s equation, solving the problem means finding a harmonic function with the prescribed boundary values.Elliptic operatorRelated: Harmonic functions are the basic solutions of the Laplace equation.Hadamard three-circle theoremRelated: Linear interpolation in log-radius comes from harmonic functions on an annulus.Jensen's formulaRelated: Away from zeros, the logarithmic modulus of a holomorphic function is harmonic.Schwarz reflection principleRelated: Real and imaginary parts of holomorphic extensions provide reflected harmonic extensions.Earnshaw's theoremNarrower topic: Electrostatic potential in empty space is harmonic, placing it under the theorem’s constraint.Borel–Carathéodory theoremRelated: The real part of every holomorphic function is harmonic, supplying the theorem’s scalar control.Morera's theoremCompared with: Harmonicity alone does not encode the complex-valued contour condition Morera requires.Harnack's principleNarrower topic: These are the functions in the sequence and the possible finite limit.Weyl's lemmaRelated: The lemma concludes that the distribution is represented by a function of this kind.
KnowraHarmonic functionLinked fromLinked fromThe 17 pages that link to Harmonic function, each with the reason it gives.All 17Broader topic 4Related 9Narrower topic 2Compared with 2Holomorphic functionCompared with: Holomorphic functions yield harmonic components, but harmonicity alone does not ensure holomorphicity.Complex analysisRelated: Real and imaginary parts of holomorphic functions are harmonic, but harmonic functions need not arise globally this way.Laplace's equationBroader topic: These functions are precisely the classical solutions of Laplace's equation.Scalar fieldRelated: Harmonic functions are scalar fields with a distinctive local averaging property.Maximum principleBroader topic: Harmonic functions are the classical solutions for which interior extrema force constancy.Cauchy–Riemann equationsRelated: The equations imply that both components of a sufficiently smooth holomorphic function are harmonic.Potential theoryBroader topic: Harmonic functions are the basic local objects characterized by potential theory.Dirichlet problemBroader topic: For Laplace’s equation, solving the problem means finding a harmonic function with the prescribed boundary values.Elliptic operatorRelated: Harmonic functions are the basic solutions of the Laplace equation.Hadamard three-circle theoremRelated: Linear interpolation in log-radius comes from harmonic functions on an annulus.Jensen's formulaRelated: Away from zeros, the logarithmic modulus of a holomorphic function is harmonic.Schwarz reflection principleRelated: Real and imaginary parts of holomorphic extensions provide reflected harmonic extensions.Earnshaw's theoremNarrower topic: Electrostatic potential in empty space is harmonic, placing it under the theorem’s constraint.Borel–Carathéodory theoremRelated: The real part of every holomorphic function is harmonic, supplying the theorem’s scalar control.Morera's theoremCompared with: Harmonicity alone does not encode the complex-valued contour condition Morera requires.Harnack's principleNarrower topic: These are the functions in the sequence and the possible finite limit.Weyl's lemmaRelated: The lemma concludes that the distribution is represented by a function of this kind.