I read that the Poisson Kernel for the upper half-plane, $K_y(x)=\frac{1}{\pi}\frac{y}{y^2+x^2}$ is an approximation to the identity. The text states this without proof and I am hoping to see a proof of this fact. Thank you in advance for the help.
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$\begingroup$ Numerator should be $y$ not $y^2$. Take a look at Dirac's $\delta$ distribution smooth approximation. Use what I explained there with $n=1$, $\zeta(x)=\frac{1}{\pi}\frac{1}{1+x^2}$ and $c=1$. $\endgroup$– peek-a-booNov 6, 2021 at 2:34
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$\begingroup$ @peek-a-boo you are right! My mistake. Also, I am having trouble following your other answer. I think it’s just a bit over my head. I corrected the error above. $\endgroup$– AbdulNov 6, 2021 at 3:31
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