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 Chapter 8 in RTR and especially branch points. 
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Joined: 07 Jun 2008, 08:21
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Post Chapter 8 in RTR and especially branch points.
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
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RTRDiscuss8.02.pdf [60.99 KiB]
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09 Mar 2009, 15:42

Joined: 11 Jul 2009, 20:45
Posts: 25
Post Re: Chapter 8 in RTR and especially branch points.
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!


07 Aug 2009, 03:34
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Post Re: Chapter 8 in RTR and especially branch points.
harried wrote:
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


11 Aug 2009, 17:40

Joined: 07 May 2009, 16:45
Posts: 62
Post Re: Chapter 8 in RTR and especially branch points.
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.


16 Aug 2009, 04:36

Joined: 03 Jul 2011, 14:43
Posts: 8
Post Re: Chapter 8 in RTR and especially branch points.
I wish to point out that there is another way to check if there is a 'branch point at infinity': one can simply take a very large circle around the origin (or any point, if one chooses to do so), and seeing if the function returns to the same value after an anti-clockwise rotation for any large enough circle.

This is useful for cases where it is not so easy to use the 'inverse mapping of infinity', but essentially gives the same results. It's also mentioned in the book.


16 Jul 2011, 04:08

Joined: 11 Jul 2009, 20:45
Posts: 25
Post Re: Chapter 8 in RTR and especially branch points.
Vasco,,

I've thought about this Cartesian map of x + iy to rE(i Theta ) and it is BS !!

I've thought about it for 5 years now ,, and ain't no way you get another dimension with every turn !

If existence is multivalued then its NON-EXISTING !!


09 Aug 2012, 18:33
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Post Re: Chapter 8 in RTR and especially branch points.
harried wrote:
Vasco,,

I've thought about this Cartesian map of x + iy to rE(i Theta ) and it is BS !!

I've thought about it for 5 years now ,, and ain't no way you get another dimension with every turn !

If existence is multivalued then its NON-EXISTING !!

Hi harried
I'm not quite sure what you are saying here. Do you agree that any point in the complex plane can be labelled as (x,y) or as (r,\theta) and that the label (r,\theta\pm2n\pi) where n=1,2,... is just as good a label as (r,\theta)?
What do you mean by BS - "bloody stupid"?
What exactly are you disagreeing with when you talk about dimensions and turns?
Cheers
Vasco


16 Aug 2012, 14:01
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