See fig 8.2 .This time four branch points are needed :
and .Make two branch cuts, one joining the point with and the other with .
The topology is again that of a torus without the above mentioned points where

What is the logic to making the cuts between these points rather than some other way? Also why in this case is there no branch point at infinity? My instinct was to make cuts between i and -i, and another from 1 to infinity and infinity and -1, yielding a torus again. I'd be interested to know more, as I'm not sure I really understand this! :-) Thanks
J

It does not matter where you make the cuts, because there is no cut at all on Riemann surface. A cut is merely an artifact that appears where you split the Riemann surface into sheets, and depends on how you make the splitting. Any of the proposed cuts in the posts above is OK, but I prefer the Zahoor's, because it is the simplest one.

In this case, what really matters is the following:

• The Riemann surface has two sheets over each point;
• There are four branch points at , , , and ;
• As we circle by one complete turn around each individual branch point, we end up in a different sheet;
• Circling again, we come back to the starting point.

As for the branch point at infinity, as we circle around a very large loop, we find that the function does not change. Therefore, there is no branch point at infinity.

Hi, I'm still struggling with the former demonstration, . My questions are:
(i) I can see that it has three branch points, 1, and , but why an 'infinity branch point'?
(ii) I couldn't derive that each branch point is of the order of 2.
(ii) The figures are quite messy. Given an order of 2, I can understand only figure a. I have no idea about all the cuts and torus.
Please help!

OK. I've managed to get one down - the answer to branch points having order 2, which does need a bit of manipulation. Can somebody please help explaining why the 'infinity branch point' and how all the 'cuts' and 'glue' and 'torus' (Fig 8.2b) work? Thanks!

Hi Variounes
I have posted a document in Exercise Discussion which I hope will answer your questions about branch points and figure 8.2 in RTR.