Gary
GaryVasco
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Post by Gary on Apr 29, 2016 5:38:01 GMT
Vasco, There are a few diagrams in the attachment that may be of interest. nh.ch5.ex25.pdf (175.07 KB) (revised 04/28/16) Gary |
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Post by Admin on Apr 29, 2016 14:53:23 GMT
Gary
Yes, interesting diagrams! Figure 1 reminds me of exercise 24 chapter 1 part (iii). One thing: In your document sentence 1 you write "In figure 1, with $f(z)<1$, the red point..." Shouldn't that be "In figure 1, with $|z|<1$, the red point...".
As you will have noticed I haven't published a solution to exercise 26. I do have a solution, but I'm not totally happy with it yet. The equation for $\mathcal{S}_{E}(z)$ half way down page 257, the derivation of which Needham sets as an exercise, holds some surprises. I won't say any more at this point, so's not to prejudice you in any way.
Vasco
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Gary
GaryVasco
Posts: 3,352
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Post by Gary on Apr 29, 2016 15:15:30 GMT
Vasco,
Yes, it should.
Regarding the exercise from Ch 1, I had not reviewed it, but it seems more familiar now. My thought was that it is hard to imagine where a complex power series might converge, or where it might go if it does not converge, because there is amplitwist in every summand. The vector graph provides at least a partial map, sometimes drawn by what at first appears to be an erratic map maker. Given much more time, it would be fun to explore different |z| and different arg(z) with these and other power series.
I peeked at 26 yesterday and decided I needed to do some more reading. It sounds interesting.
Gary
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Post by Admin on Apr 30, 2016 16:04:22 GMT
Gary Looking at your figure 2 it looks as though you could get some interesting patterns by experimenting with different values as you say. You can already see a pattern emerging after just a few steps. It reminded me of Turtle Graphics, which I used on the BBC micro in the 1980s, and the associated programming language Logo which I also used. You can download a free compiler/interpreter for Windows-based computers called MSWlogo, I also used a wonderful book called Turtle Geometry - the Computer as a Medium for Exploring Mathematics by Harold Abelson and Andrea diSessa published by the MIT Press. I would recommend it if you can get hold of a copy. It starts off with quite simple stuff and then goes on to some quite advanced stuff later in the book and even touches on relativity. The reason it's called Turtle graphics is because Logo imagines a turtle to which you give commands: Take 2 steps, turn through 120 degrees, take 4 steps etc. Its very good for vector graphics. Vasco |
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Gary
GaryVasco
Posts: 3,352
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Post by Gary on Apr 30, 2016 20:37:06 GMT
Vasco,
I have heard of Logo and Turtle graphics and seen it recommended. I hesitate to adopt yet another programming system because each one has it's own peculiarities that require an expenditure of time to gain proficiency. I'm just starting to feel comfortable with Mathematica. But I will look for the book and perhaps give it a try if there is a version that works with OS X. Some of it might be transferable.
Gary
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Post by Admin on Apr 30, 2016 20:45:19 GMT
Gary I didn't mean to suggest using it for the work on Needham's book. Just as a separate project if you have any free time  . Vasco |
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Gary
GaryVasco
Posts: 3,352
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Post by Gary on Apr 30, 2016 21:05:13 GMT
Gary I didn't mean to suggest using it for the work on Needham's book. Just as a separate project if you have any free time . Vasco Vasco, Right now I'm viewing Cedric Villani discussing Landau damping, so I probably have time for Turtle Geometry. Gary |
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Post by Admin on May 1, 2016 6:38:58 GMT
Gary
Funnily enough that rings a bell. I used to work in the theoretical physics department at Culham Laboratory (part of the UKAEA) created to do research into Plasma Physics with a view to obtaining energy from fusion, and I remember Landau damping being discussed. I know nothing about it myself, I just remember the phrase. I suppose it's one way to use up your spare moments!
Vasco
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