= Solution
{c}
For the potential strip, exponentiation followed by a <Möbius transformation> gives
$$
\zeta=-\frac1{1+e^{\pi w/a}}.
$$
Since $e^{\pi w/a}$ lies in the upper half-plane, so does $\zeta$. The map sends $A,E$ to $-1$, $D=ia$ to infinity, and $B,C$ to zero.
For the first-quadrant <hodograph plane>, successively square $q$, apply $(q^2-1)/(q^2+1)$, and then use a negative square:
$$
\zeta=-\left(\frac{q^2-1}{q^2+1}\right)^2.
$$
The first map takes the quarter disk into the upper half of the unit disk. The fractional transformation takes this to the second quadrant, whose negative square is the upper half-plane. The boundary points $0,i,1$ map to $-1,\infty,0$, as required. Uniqueness of the normalized <conformal map> therefore identifies the two expressions:
$$
\boxed{\frac1{e^{\pi w/a}+1}
=\left(\frac{(w')^2-1}{(w')^2+1}\right)^2.}
$$
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