The integral kernel in the Poisson integral, given by
(1)
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for the open unit disk . Writing and taking gives
(2)
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(3)
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(4)
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(5)
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(6)
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(Krantz 1999, p. 93).
In three dimensions,
(7)
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where and
(8)
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The Poisson kernel for the -ball is
(9)
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where is the outward normal derivative at point on a unit -sphere and
(10)
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Let be harmonic on a neighborhood of the closed unit disk , then the reproducing property of the Poisson kernel states that for ,
(11)
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(Krantz 1999, p. 94).