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With the usual exterior-boundary interpretation that the harmonic function remains bounded as $r\to \infty$, positive radial powers and $\log r$ are excluded. HenceThe series must converge locally for $r>1$, with any required finite boundary values at $r=1$. These are the interior harmonic Fourier expansions in planar concentric domains after the inversion $r\mapsto 1/r$.

If “finite” means only finite at each finite point, without a condition at infinity, growing solutions such as $r\cos \theta$ and $\log r$ are allowed. Under that literal reading the general exterior expansion iswith local convergence. Boundedness at infinity is the condition selecting the decaying exterior expansion.

Use separation of variables in Laplace's equation: writing $\varphi =R(r)\Theta (\theta )$ givesSingle-valuedness requires $2\pi$-periodicity, so the nonconstant angular Fourier modes have integer $n\ge 1$ and are $\cos n\theta ,\sin n\theta$. Their radial solutions are $r^n,r^{-n}$. The zero angular mode has radial solutions $1,\log r$. These give the harmonic Fourier expansions in planar concentric domains used below.

For the annulus with $0<a<b$, constant boundary data suggest a radial harmonic function $A+B\log r$. Enforcing both boundary values givesThis has the required boundary values and satisfies $(r{\varphi }_r{)}_r=0$. It is unique: the difference of two continuous solutions is harmonic in the annulus and zero on both boundary circles, so the maximum principle for harmonic functions, applied to the difference and its negative, makes it zero.

Set $\varphi =R(r)\Theta (\theta )$ in Laplace's equation. Separation gives ${\Theta }^{\prime \prime }+n^2\Theta =0$ and $r^2{R}^{\prime \prime }+r{R}^{\prime }-n^{2}R=0$. Single-valued angular periodicity requires integer $n\ge 0$. For $n>0$ the radial factors are $r^n,r^{-n}$; for $n=0$ they are $1,\ln r$. A term linear in $\theta$ is excluded by periodicity. Thus the harmonic Fourier expansions in planar concentric domains have the general formFor $0<a<b$, the specified boundaries contain only a constant mode and a first cosine mode. The two-boundary annular harmonic interpolation givesAt $r=a$, the cosine term vanishes and the logarithmic term is $U$; at $r=b$, the first term vanishes and the cosine amplitude is $V$. Each component is harmonic, and the maximum principle ensures uniqueness for these continuous Dirichlet boundary conditions.