= Solution
For <separation of variables>, let $\Phi=S(\sigma)T(\tau)$. The <Laplace equation> gives
$$
-\frac{S''}{S}=\frac{T''}{T}=\lambda.
$$
A nonzero $2\pi$-periodic angular solution requires $\lambda=n^2$, $n=0,1,\ldots$. For $n\geq1$, $S$ is a linear combination of $\cos(n\sigma)$ and $\sin(n\sigma)$, while $T''-n^2T=0$ gives hyperbolic sine and cosine. For $n=0$, periodicity makes $S$ constant, and $T=A_0+B_0\tau$. Negative separation constants would give nonperiodic real exponentials in $\sigma$.
Superposition yields
$$
\boxed{\begin{aligned}
\Phi(\sigma,\tau)={}&A_0+B_0\tau\\
&+\sum_{n=1}^\infty\Big\{[A_n\cosh(n\tau)+B_n\sinh(n\tau)]\cos(n\sigma)\\
&\hspace{40mm}+[C_n\cosh(n\tau)+D_n\sinh(n\tau)]\sin(n\sigma)\Big\}.
\end{aligned}}
$$
Completeness follows by expanding a <harmonic function> in a <Fourier series> in the periodic coordinate: its <Fourier coefficients> must satisfy exactly these radial equations. As usual, the coefficients are restricted by the convergence required of the series on compact subregions. The <bipolar coordinates> domain imposes no boundary condition at $\tau=0$ or $\tau=\infty$, so both hyperbolic branches remain available.
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