|
|
Archived: 07 Aug 2014, 09:45
|
Author |
Message |
timray60
Joined: 12 Nov 2008, 02:55 Posts: 12 Location: Japan
|
 Exercise 8.7
I am hopeful that I am on the right track for what is being asked to verify in exercise 8.7. I'm not 100% sure of the accuracy though. See the insert. Thanks Tim
|
22 Nov 2008, 12:48 |
|
 |
tonychinnery
Joined: 16 Jul 2010, 07:42 Posts: 11 Location: Florence, Italy
|
 Re: Exercise 8.7
If the transformation of the Riemann sphere given by those equations is supposed to be a 90° rotation around the line joining the points +1 and -1 , then these two points should remain invariant, but I am stuck allready because when z=1, t becomes 0. Evidently there is something essential I am missing here. Where am I wrong?
|
04 Aug 2010, 07:55 |
|
 |
tonychinnery
Joined: 16 Jul 2010, 07:42 Posts: 11 Location: Florence, Italy
|
 Re: Exercise 8.7
Perhaps it is enough to show that z = (i-t)/(i+t) maps the real numbers to the unit circle. Well, we know the modulus of z is 1 when t is real (the moduli of the upper and lower bits are the same) and we are told that the number line is mapped to a circle under a bilinear mapping. We know that the circle passes through i, 1, and -i (by substituting 1, 0 and -1 for t ) so therefore it must be the unit circle. Well, its supposed to be an easy problem!
|
06 Aug 2010, 20:37 |
|
 |
vasco
Supporter
Joined: 07 Jun 2008, 08:21 Posts: 235
|
 Re: Exercise 8.7
Tony This exercise is not marked as simple in my book. It has the symbol "needs a bit of thought" next to it. Have you looked at my solution? Here is the link to it, 8.7b http://www.roadtoreality.info/viewtopic.php?f=19&t=286
|
08 Aug 2010, 16:05 |
|
 |
|
|