= Dirichlet problem between bipolar coordinate circles
{c}
For a <harmonic function> between $\tau=a$ and $\tau=b$, $0<a<b$, in <bipolar coordinates>, suppose $\Phi(\sigma,a)=0$ and $\Phi(\sigma,b)=F(\sigma)$. Write the periodic boundary values as a <Fourier series>
$$
F(\sigma)=F_0+\sum_{n\geq1}[A_n\cos(n\sigma)+B_n\sin(n\sigma)].
$$
<Separation of variables> gives
$$
\Phi(\sigma,\tau)=F_0\frac{\tau-a}{b-a}+\sum_{n\geq1}\frac{\sinh[n(\tau-a)]}{\sinh[n(b-a)]}[A_n\cos(n\sigma)+B_n\sin(n\sigma)].
$$
Each radial factor solves $M''-n^2M=0$ and has endpoint values zero and one; the mean mode solves $M''=0$. Sufficiently regular boundary data give the required <Fourier series> solution. <Uniqueness> follows from the <maximum principle> for a <harmonic function> with zero <Dirichlet boundary conditions> on both circles. The formula follows the coordinate values; $\tau=b$ is the geometrically smaller circle.
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