= Solution
{c}
On the free streamline choose $q=e^{-i\theta}$ with $-\pi/2\leq\theta<0$. The lower endpoint of this parameter interval is $D$, and $\theta\to0^-$ approaches $C$. The fractional expression in the map is
$$
\frac{e^{-2i\theta}-1}{e^{-2i\theta}+1}=-i\tan\theta.
$$
Hence $e^{\pi w/a}=-\csc^2\theta$, and the branch on the upper edge of the potential strip is
$$
w=\frac a\pi\log(\csc^2\theta)+ia.
$$
Differentiate to obtain $dw/d\theta=-2a\cot\theta/\pi$. Since $dw/dz=q$, the <chain rule> gives
$$
\boxed{\frac{dz}{d\theta}=\frac1q\frac{dw}{d\theta}
=-\frac{2ae^{i\theta}\cot\theta}{\pi}.}
$$
The sign and interval are consistent with the free surface moving to the right and downward from $D$.
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