deant wrote:

Is Penrose's formula wrong here? Should it be tan, rather than cot?

According to the standard polar coordinates as shown in Fig. 22.16 (p.563), theta=0 corresponds to the (x,y,z) point (0,0,1) - the North pole of the Riemann sphere.

Normally in stereographic projection, the north pole of the Riemann sphere maps to the origin of the complex plane. That's what you get if you use tan.

Using cot gives you a 'reflected' stereographic projection, where the south pole of the Riemann sphere maps to the origin, and the North pole to infinity.

Perhaps that's the convention in general relativity theory, but since it contradicts the conventions established earlier in the book, I'd have expected Penrose to mention it if that were the case...?

In this exercise (18.11), Penrose is using the convention where the stereographic projection is from the North pole.

On page 144 paragraph 8.3 he is using projection from the south pole in figure 8.7(a) and from the north pole in figure 8.7(b).

Projection from the north pole gives the result you are asked to prove in exercise 18.11 -

exp{i\phi}cot(\theta/2) - and since these two projections are the inverse of each other (see figure 8.7 referred to above) the projection from the south pole is

exp{-i\phi}tan(\theta/2).

Another potentially confusing thing is that in America the roles of \theta and \phi are reversed in the definition of spherical polar coordinates. The American convention has the advantage that \theta remains in the complex plane and so coincides with the normal notation for complex numbers. Penrose is using the UK convention for spherical polar coordinates in this exercise and on page 563 figure 22.16.

See Visual Complex Analysis by Tristan Needham chapter 3 section IV paragraph 5 where the American convention is used by a UK author for the reasons I gave above.

So to sum up Penrose is not being inconsistent, since on page 144 he talks about both projections, but I think it would be less confusing if authors stated which conventions they are using when quoting formulae.

There are two conventions involved here - the spherical polar coordinate convention and the north or south pole convention.