Harmonic Function Theory/Definitions, elementaries

< Harmonic Function Theory

Definition (harmonic function):

A harmonic function is a function $f\in {\mathcal {C}}^{2}(\mathbb {R} ^{n})$ such that $\Delta f=0$ .^{[note 1]}

Proposition (closure properties of harmonic functions):

Harmonic functions are closed under the following operations:

Addition, subtraction
Locally uniform convergence

Footnotes

↑ The Laplace operator $\Delta$ is defined as $\Delta f:=\sum _{k=1}^{n}{\frac {\partial ^{2}}{\partial x_{k}^{2}}}f$ .

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