Hi everyone!
I don't know if I am the only one, but I am finding Chapter 8 of RTR to be very difficult. There are lots of new concepts which are not explained in enough detail for me and as a result I have given this chapter a lot of thought and hard study because looking forward in the book it looks as though understanding chapter 8 could be crucial to understanding later chapters.
Member Variounes has also expressed his frustration in trying to understand parts of this chapter.
I have attached to this post a document which is the result of all my studying and hard work. It may not be without flaws, but I hope that it will help others to understand this chapter and may be used as a document to stimulate further discussion of the difficult but very interesting topics covered in chapter 8.
Good Luck and please let me have any comments/corrections or feedback so that we can all get something useful out of this. Thanks
Vasco

Hi Vasco, I've read Penrose's discussion of the mapping from Z= X+ iY to the mulivalued Z = sqrt(X*2+Y*2)*exp(arctan(Y/X),,and Roger only casually mentions the ambiguity of going to and fro, except to say "IT INTRODUCES AMBIGUITIES"!
The whole idea of mapping the X+iY plane into a "SPIRAL STAIRCASE" of (R, iTheta), is treated rather slipshod! Many ambiguities, exp(1) = exp(1+2*PI) arise from this mapping ( see figs 6.1, and 8.1 )!
Roger has bypassed an explanation of why this mapping gains so much for us!

Harried
I think you really mean Z = sqrt(X**2+Y**2)*exp(i * arctan(Y/X)) and
exp(1) = exp(1+2*PI*i)
don't you?
I disagree with you that Penrose does not discuss the advantages of the Riemann surface view of log z and other multi-valued functions. Have you read and understood section 8.1?
Vasco

Quoting your document:

1. Some say that if you need to go round the origin n times before the function returns to its original chosen value, then the order is n-1 and use it as a measure of how many more times you need to go round the branch point to get back to the same function value than you would for a single valued function. With this definition, a ’normal’ function (single-valued) would have branch points of order 0 (no branch points).

2. Others say that if you need to go round the origin n times before the function returns to its original chosen value, then the order is n. With this definition, a ’normal’ function (single-valued) would have branch points of order 1 (no branch points). One advantage of this definition is that the number of sheets required to construct the Riemann surface is also n. In RTR Penrose uses this definition and so in the rest of this discussion I will use his definition to avoid confusion for people reading RTR, which is most of us.



If I understand correctly, Number 1 Is the way Tristan Needham's Visual Complex Analysis gives the definition. Strange, because Needham is a student of Penrose, and that book is dedicated to Penrose.

I think your document is good, by the way.