= Harmonic expansion on a disk
{title2=$u(r,\theta)=a_0+\sum_{n\geq1}r^n(a_n\cos(n\theta)+b_n\sin(n\theta))$}
A real <harmonic function> on the unit <disk> has the displayed expansion, convergent locally uniformly with its <derivatives> in the interior. If its continuous boundary data have <Fourier coefficients> $A_0,A_n,B_n$ in the convention $f(\theta)=A_0/2+\sum_{n\geq1}(A_n\cos(n\theta)+B_n\sin(n\theta))$, its <harmonic extension> has $a_0=A_0/2$, $a_n=A_n$, $b_n=B_n$.
To obtain the radial factors, <separation of variables> in the <Laplace equation in polar coordinates> gives $r^2R''+rR'-\lambda R=0$ and $\Theta''+\lambda\Theta=0$. Angular <periodicity> restricts nonconstant modes to $\lambda=n^2$ for positive integers $n$. The radial factors are $r^n,r^{-n}$, with $1,\log r$ for the constant angular mode. Boundedness at the origin removes $r^{-n}$ and $\log r$. Each surviving factor $r^n\cos(n\theta)$ or $r^n\sin(n\theta)$ is a <harmonic polynomial>, the real or imaginary part of $(x+iy)^n$. The expansion can also be derived from the <Poisson integral on the unit disk> for the <disk>.
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