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

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?

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

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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