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Exercise 5.9 Equi-angular spiral
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Joined: 25 Mar 2008, 18:45
Posts: 2
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?

25 Mar 2008, 19:46

Joined: 13 Mar 2008, 14:06
Posts: 42
Location: Ithaca NY
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

25 Mar 2008, 21:58

Joined: 25 Mar 2008, 18:45
Posts: 2
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.

27 Mar 2008, 17:45
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