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Archived: 07 Aug 2014, 09:49
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Exercise 5.9 Equi-angular spiral
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jmc2000
Joined: 25 Mar 2008, 18:45 Posts: 2
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 Exercise 5.9 Equi-angular spiral
On page 97 of The Road To Reality by Penrose he has Fig 5.9: ........+........................................... ....................+............................... .....................*...+..........*............. ..............*................................*... ...........*..................+.................... .........*.......................................... ........*........................................... .......*............................+.............. .................................................... .................................................... ......*............................................ ......................................+............ ......*..........*.*.............................. ...............*......*...........+............... .......*........*+*..*.......+.................. ..................................................... .......................+.*.+....................... ...........*...........*............................ ..............*.....*.............................. ..................................................... So that + and * both start out from the origin, anti and clockwise respectively
With the figure it says:
"The different values of W^z ( = e^(zlogw) ). Any integer multiple of 2ipi can be added to logw, which multiplies or divides w^z by e^(z 2ipi) an integer number of times. In the general case, these are represented in the complex plane as the intersection of two equiangular spirals (each making a constant angle with straight lines through the origin).
Exercise 5.9 asks to show this.
How?
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25 Mar 2008, 19:46 |
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fallingup
Joined: 13 Mar 2008, 14:06 Posts: 42 Location: Ithaca NY
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 Re: Exercise 5.9 Equi-angular spiral
I did that one. I don't remember the details, but it involved solving the differential equation that expresses that it makes a constant angle with a line from the origin. It was an exponential spiral of some sort. Laura
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25 Mar 2008, 21:58 |
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jmc2000
Joined: 25 Mar 2008, 18:45 Posts: 2
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 Re: Exercise 5.9 Equi-angular spiral
I don't think it's correct to prove it using the solution of a differential equation because differentiation isn't covered until chapter 6.
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27 Mar 2008, 17:45 |
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