Poisson Kernel

The integral kernel in the Poisson integral, given by

for the open unit disk D(0,1). Writing z_0=re^(itheta) and taking D(0,R) gives

(Krantz 1999, p. 93).

In three dimensions,

where a=|y| and

The Poisson kernel for the n-ball is

where D_(n) is the outward normal derivative at point z on a unit n-sphere and

Let u be harmonic on a neighborhood of the closed unit disk D^_(0,1), then the reproducing property of the Poisson kernel states that for z in D(0,1),

(Krantz 1999, p. 94).

See also

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References

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Cite this as:

Weisstein, Eric W. "Poisson Kernel." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PoissonKernel.html

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