= Solution
Use the <complex logarithm> with $0\leq\arg w\leq\pi/2$ on the nonzero points of the first-quadrant square. Put $z=x+iy=\log|w|+i\arg w$. The square bounds $\operatorname{Re}w,\operatorname{Im}w\leq a$ give
$$
\boxed{0\leq y\leq\frac\pi2,\qquad x\leq\log a-\log\max(\cos y,\sin y).}
$$
The two edges meeting at $w=0$ map to the horizontal half-lines $y=0$ and $y=\pi/2$, extending indefinitely to the left. The other edges map to $x=\log a-\log\cos y$ for $0\leq y\leq\pi/4$ and $x=\log a-\log\sin y$ for $\pi/4\leq y\leq\pi/2$. They meet at $\log(a\sqrt2)+i\pi/4$.
\b[The angles are preserved at $a$, $a(1+i)$ and $ia$.] There $d\log w/dw=1/w$ is finite and nonzero. The origin is a <branch point>, has no finite logarithmic image, and supplies no fourth finite vertex at which angle preservation could be asserted.
\Image[/past-exam-of-the-short-options-paper-shared-by-prelims-and-parts-a-and-b-of-the-physics-course-of-the-university-of-oxford/2015-s1-conformal-square-maps.png]
{title=A square, its exponential annular sector, and its logarithmic image extending to the left}
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