is harmonic in , and suppose . If , then

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I like this theorem, it is beautiful and concise. The proof was kind of trivial if apply Poisson Kernel bravely. Let’s make it more general.

Set , thus define , if we are given that , then we are going to prove:

consider unit vector , by Poisson Integral,

,

where

,

since , thus we assume , we know that is homogeneous, hence

.

which is

The inner integral is .

Done.

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It seems that this conclusion cannot be generalized. because the Poisson Kernel only involves two surfaces, not N surfaces.