= Area norm of the Poisson extension in the unit disk
{title2=$\int_D|u|^2\,dA=\pi|a_0|^2/4+(\pi/2)\sum_{n\ge1}(|a_n|^2+|b_n|^2)/(n+1)$}
For square-integrable periodic boundary data and the <Fourier coefficients> used in the <Poisson kernel for the unit disk>, the displayed identity gives the squared area norm of its Poisson extension $u$. At fixed radius $r$, <Parseval's identity> gives
$$
\int_{-\pi}^{\pi}|u(r,\theta)|^2\,d\theta
=\pi\left[\frac{|a_0|^2}{2}+\sum_{n=1}^\infty r^{2n}(|a_n|^2+|b_n|^2)\right].
$$
Multiply by the <polar coordinates> <area element> factor $r\,dr$ and integrate from zero to one. The constant term contributes $\pi|a_0|^2/4$, and $\int_0^1r^{2n+1}\,dr=1/[2(n+1)]$. Interchanging the nonnegative sum and radial <integral> is justified by the <monotone convergence theorem>. The resulting factor $1/(n+1)$ shows how higher angular frequencies contribute less to the interior area norm.
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