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mathematical physics > special function > harmonic function

mathematical analysis > functional analysis > special function > harmonic function

Preferred term

harmonic function

Definition(s)

In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function ${\\displaystyle f:U\ o \\mathbb {R} ,}$ where $U$ is an open subset of ${\\displaystyle \\mathbb {R} ^{n},}$ that satisfies Laplace's equation, that is,

${\\displaystyle {\ rac {\\partial ^{2}f}{\\partial x_{1}^{2}}}+{\ rac {\\partial ^{2}f}{\\partial x_{2}^{2}}}+\\cdots +{\ rac {\\partial ^{2}f}{\\partial x_{n}^{2}}}=0}$

everywhere on $U$ . This is usually written as

${\\displaystyle \ abla ^{2}f=0}$

or

${\\displaystyle \\Delta f=0}$

Broader concept(s)

special function

Narrower concept(s)

Related concept(s)

harmonic measure

In other languages

fonction harmonique

French

URI

http://data.loterre.fr/ark:/67375/PSR-H7WG59FS-L

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