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Let $$K(x,y) = \frac{2x_n}{n \alpha(n)} \frac{1}{|x-y|^n}$$ be the Poisson Kernel, where $x \in \mathbb{R}_+^n$ (the upper half-space of in $\mathbb{R}^n$), $y \in \mathbb{R}^n$, and $\alpha(n)$ is the volume of the $n$-dimensional unit ball.

How do you show that $$\int_{\partial \mathbb{R}_+^n} K(x,y) dy =1?$$

I tried doing this in simple cases (e.g. two dimensions), and it can out pretty cleanly (I think you can also probably use complex analysis if we're in two-dimensions?). However, I couldn't figure how to solve it in general $n$ dimensions, because the exponent to the $n$-th power was giving me trouble. How would one go about showing the general case?

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    Hm, I guess I did mean space instead of plane. However, the formula is exactly what was written in Evans's Partial Differential Equations book, and he defined it as the Poisson formula on the upper half-space..2011-08-30

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