Vasco
GaryVasco
Posts: 14
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Post by Vasco on Nov 5, 2015 17:03:50 GMT
Gary
I changed my mind about my last post and decided to stick with what Needham says on p. 210 in the paragraph under the C-R equations (6):
"...as with the underlying amplitwist concept, these equations must be satisfied throughout an infinitesimal neighbourhood of a point in order that the mapping be analytic there [see Exercise 12]."
This points us to exercise 12 which is presumably supposed to make this clear, but at the moment seems to be just causing confusion. I shall put my thinking cap on and see if I can find a way to summarise it all in an understandable way.
Vasco
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Vasco
GaryVasco
Posts: 14
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Post by Vasco on Nov 5, 2015 17:04:13 GMT
Gary I have had a rethink and I have amended both versions of exercise 12 and published these amended versions. Most of the changes are to part (ii). My main point is that because in part (ii) of exercise 12 we deduce that the C-R equations are satisfied at the isolated point \(z=0\) (the origin), we cannot use (7) and (8) to deduce the value of \(f '(0)\) for the reasons given in the paragraph on page 210 just below the equations (6). There is a better explanation (I hope) in the solutions themselves. See what you think. Vasco
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Gary
GaryVasco
Posts: 3,352
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Post by Gary on Nov 6, 2015 6:00:01 GMT
Vasco,
Vasco,
I am struggling a bit with the format. In previous forums I have used, any user may begin a thread. Here it appears that a thread exists for each chapter and all user input is via replies. At least, I don't see a way to initiate a thread or a new comment other than a reply.
I followed the MathJax links, but could not find a list of the codes. (I later realized that the codes are just Tex or Latex.)
I think I already replied to your latest version of 4:12 on the blog to the effect that I agree with most of it. I had one additional thought regarding part (ii) and the statement on p. 210 that the CR "equations must be satisfied throughout an infinitesimal neighborhood of a point in order for the mapping to be analytic there". If the equations are satisfied at z = 0, and if 0 can be regarded as a point and region disconnected from other points on the plane, then 0 itself is an infinitesimal neighborhood of zero extent. If the CR equations are satisfied there, then f(0) is analytic by the rule from Ex. 7 and by (7) and (8). This seems to me to be a point of divergence of the natural geometric concept of continuity according to Newton and Needham, which would classify it as non-analytic, in contrast to the Weierstrass arithmetic definition by closeness of points in the range, which would classify it as analytic.
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Gary
GaryVasco
Posts: 3,352
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Post by Gary on Nov 6, 2015 6:14:20 GMT
Two more points I forgot to include in previous. The forum buttons and links respond very slowly compared to the blog.
I hope you will leave the exercise solutions on the old Google site, as they can be accessed much more quickly and conveniently there.
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Gary
GaryVasco
Posts: 3,352
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Post by Gary on Nov 6, 2015 9:33:46 GMT
Vasco, I, too, have completely revised my version of 12, but I found I had no need to apply my thoughts about 0 as a disconnected region of its own. It now appears to me that the axes are continuous at 0. Still, my answer is a little bit different. This is a very curious function. Gary Attachments:nh.ch4.ex12.pdf (131.82 KB)
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Vasco
GaryVasco
Posts: 14
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Post by Vasco on Nov 8, 2015 14:27:08 GMT
Gary
I have rewritten for a second time my solutions to both versions of exercise 12 chapter 4 to try and make the use of equations (7) and (8) on page 210 as clear as possible. The way they are presented in the book can easily lead to confusion and as a result they can be used in inappropriate circumstances.
I have tried to make their use as clear as possible in the published solutions.
Thanks for your helpful comments over the last couple of weeks working on chapter 4. They have enabled me to improve on my original solutions - I think!!
Vasco
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