= Poisson kernel for the unit disk
{c}
{title2=$P_r(\theta)=(1-r^2)/(1-2r\cos\theta+r^2)$}
For $0\le r<1$, this <Poisson kernel for the unit disk> has the uniformly convergent <Fourier series>
$$
P_r(\theta)=1+2\sum_{n=1}^\infty r^n\cos(n\theta).
$$
The identity follows by taking the <real part> of the <geometric series> $1+2\sum_{n\ge1}(re^{i\theta})^n$. For integrable $2\pi$-periodic boundary data $f$, its Poisson extension is
$$
u(r,\theta)=\frac1{2\pi}\int_{-\pi}^{\pi}P_r(\theta-t)f(t)\,dt
=\frac{a_0}{2}+\sum_{n=1}^\infty r^n[a_n\cos(n\theta)+b_n\sin(n\theta)],
$$
where $a_n,b_n$ are the usual <Fourier coefficients>. Each term is a <harmonic function> in the open <disk>. For continuous boundary data, the extension solves the <Dirichlet problem> with those boundary values.
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