= Solution
{c}
The <complex exponential function> gives the natural onto map:
$$
\boxed{w=e^z,\qquad e^{1/4}<|w|<e^{1/2}.}
$$
Its derivative is never zero, so it preserves angles locally. Conversely any point in this <annulus> has a logarithm with real part in $(1/4,1/2)$, proving surjectivity.
There is a terminology qualification. This map is not injective, since $e^{z+2\pi i}=e^z$ and the strip contains both points. If “conformal mapping onto” requires a <biholomorphism>, no such map exists: the strip is convex and <simply connected space>, whereas the <annulus> contains a loop of <winding number> one around zero that cannot contract inside it. A <biholomorphism> would be a <homeomorphism> and preserve this property.
Thus \b[the <complex exponential function> is a locally conformal covering; a <biholomorphism> is impossible]. It identifies imaginary translates by $2\pi i$. The printed request is satisfied under its local-angle-preserving convention, with the injectivity distinction made explicit. This is <exponential covering of an annulus by a vertical strip>.
\Image[/past-exam-of-the-algebra-analysis-and-differential-equations-1-course-of-the-university-of-oxford/2007-exponential-strip.png]
{title=The strip wraps repeatedly around an <annulus> under the <complex exponential function>}
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