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Mean-Value Property

Let a function h:U->R be continuous on an open set U subset= C. Then h is said to have the epsilon_(z_0)-property if, for each z_0 in U, there exists an epsilon_(z_0)>0 such that D^_(z_0,epsilon_(z_0)) subset= U, where D^_ is a closed disk, and for every 0<epsilon<epsilon_(z_0),

h(z_0)=1/(2pi)int_0^(2pi)h(z_0+epsilone^(itheta))dtheta.

If h has the mean-value property, then h is harmonic.

See also

Harmonic Function

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References

Krantz, S. G. "The Mean Value Property on Circles." §7.4.1 in Handbook of Complex Variables. Boston, MA: Birkhäuser, p. 94, 1999.

Referenced on Wolfram|Alpha

Mean-Value Property

Cite this as:

Weisstein, Eric W. "Mean-Value Property." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Mean-ValueProperty.html

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