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[parent] Gauss' mean value theorem for harmonic functions (Theorem)

If the function $u(z) \equiv u(x,\,y)$ is harmonic in a domain of complex plane which contains the disc $|z-z_0| \leqq r$ , then

$\displaystyle u(z_0) = \frac{1}{2\pi}\int_0^{2\pi}u(z_0+re^{i\varphi})\,d\varphi.$ (1)

Conversely, if a real function $u(x,\,y)$ is continuous in a domain $G$ of $\mathbb{R}^2$ and satisfies on all circles of $G$ the equation (1), then it is harmonic.



"Gauss' mean value theorem for harmonic functions" is owned by PrimeFan. [ owner history (2) ]
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See Also: Gauss' mean value theorem


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Cross-references: equation, circles, continuous, real function, conversely, disc, contains, complex plane, domain, harmonic, function

This is version 5 of Gauss' mean value theorem for harmonic functions, born on 2005-01-22, modified 2005-01-22.
Object id is 6658, canonical name is GaussMeanValueTheoremForHarmonicFunctions.
Accessed 3809 times total.

Classification:
AMS MSC: 31A05 (Potential theory :: Two-dimensional theory :: Harmonic, subharmonic, superharmonic functions)
 30F15 (Functions of a complex variable :: Riemann surfaces :: Harmonic functions on Riemann surfaces)
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