Solution
= Solution
If $f_0(z)=i\operatorname{Log}z$ is the projection from part (b), then any other Mercator projection $f$ gives an automorphism $f\circ f_0^{-1}$ of $E$ preserving its two ends. Part (c) therefore yields
$$
\boxed{f(z)=i\operatorname{Log}z+ic\qquad(c\in\mathbb R)}.
$$
Every map in this family plainly has the required limiting behaviour.