= Solution
{c}
For a nonzero separable solution of the <Laplace equation in polar coordinates>, write $T(r,\theta)=R(r)\Theta(\theta)$ with $r>0$. Multiplying the <partial differential equation> by $r^2$ and dividing where $R\Theta\ne0$ gives
$$
\frac{r^2R''+rR'}R=-\frac{\Theta''}{\Theta}=\lambda,
$$
where $\lambda$ is a real <separation constant>. The resulting <ordinary differential equations> are
$$
r^2R''+rR'-\lambda R=0,\qquad\Theta''+\lambda\Theta=0.
$$
The equations continue across zeros of the factors by continuity. The radial equation is a <Cauchy-Euler equation>; putting $s=\log r$ changes it to $d^2R/ds^2-\lambda R=0$.
For \b[positive separation constant], write $\lambda=k^2$ with $k>0$. The radial factors are powers, while the angular factors are <sine> and <cosine>:
$$
\boxed{T=(A r^k+B r^{-k})(C\cos(k\theta)+D\sin(k\theta))}.
$$
For \b[zero separation constant], each transformed equation has zero second <derivative>, giving
$$
\boxed{T=(A+B\log r)(C+D\theta)}.
$$
For \b[negative separation constant], write $\lambda=-k^2$ with $k>0$. The radial factors oscillate in $\log r$, and the angular factors are <exponential functions>:
$$
\boxed{T=(A\cos(k\log r)+B\sin(k\log r))(C e^{k\theta}+D e^{-k\theta})}.
$$
These exhaust the nonzero real separated solutions before imposing angular <periodicity> or regularity at the origin. The zero solution is also allowed. Since <Laplace's equation> is linear, sums of these separated solutions are solutions wherever the sums and required <derivatives> converge appropriately.
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