Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2016 / S1 / 2

A conformal map preserves local oriented angles and has a nonzero local scale factor. For an orientation-preserving complex map this is equivalent to holomorphy with $f^{\prime }(z)\ne 0$ at the point concerned. Infinitesimal figures are rotated and uniformly scaled there, but finite shapes need not be preserved: derivative magnitude and direction can vary across a region. Antiholomorphic maps preserve unoriented angles while reversing orientation; the convention being used must be stated.

For real $x\ne -1$, $w=1-2/(x+1)$ is real. Its finite image is the entire real axis except $w=1$, which is approached only as $x\to \pm \infty$. The point $x=-1$ maps to infinity. On the extended complex plane the extended real axis maps onto itself, with $z=\infty$ mapping to one.

The identity $w-1=-2/(z+1)$ gives $\boxed{|w-1|=2/r}$ on $|z+1|=r$. Thus the image circle has center one and radius $2/r$. Points strictly inside the original circle satisfy $|w-1|>2/r$; the center itself maps to infinity. The interior therefore becomes the exterior together with infinity on the Riemann sphere, rather than the interior of the image circle. This inversion is a useful warning against assuming conformality preserves global inside/outside assignments.

For $z=e^{i\theta }$, $w=i\tan (\theta /2)$, which is purely imaginary; $z=-1$ supplies the point at infinity. Conversely, the inverse map is $z=(1+w)/(1-w)$. Requiring $|w|=1$ is equivalent to $|z-1|=|z+1|$, hence $\boxed{\mathrm{Re}z=0}$, including the appropriate infinite point in the extended-plane formulation. The unit circle and imaginary axis are exchanged, but the real axis is not.