= Solution
Write $s=\log r$, so the two circular boundaries are $s=0$ and $s=2$. The zero angular mode must interpolate between $-1$ and $1$, giving $s-1=\log r-1$.
For the $n$th cosine mode, the radial factor has equal value $A_n$ at both boundaries. The unique <harmonic function> with those data is
$$
A_n\frac{\cosh(n(s-1))}{\cosh n}\cos(n\theta).
$$
Hence the solution of the annular <Dirichlet problem> is
$$
\boxed{
\phi(r,\theta)
=\log r-1
+\sum_{n=1}^{\infty}
A_n\frac{\cosh(n(\log r-1))}{\cosh n}\cos(n\theta)
}.
$$
At $r=1$ and $r=e^2$, the <hyperbolic cosine> quotient equals one, so the <boundary conditions> are satisfied term by term.
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