Home_7 - Homework 7 Stephen Taylor June 4, 2005 Pages 54-...

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Unformatted text preview: Homework 7 Stephen Taylor June 4, 2005 Pages 54- 57 : 6. Evaluate the following cross ratios. We first note that the cross ratio is defined to be Definition 1. If z 1 ∈ C then ( z 1 , z 2 , z 3 , z 4 ) is the image of z 1 under the M¨ obius transformation which takes z 2 → 1, z 3 → 0, z 4 → ∞ where the transformation is given by: S ( z ) = z- z 3 z- z 4 · z 2- z 4 z 2- z 3 (c) (0 , 1 , i,- 1) S (0) =- i 0 + 1 · 1 + 1 1- i = 1 + i (d) ( i- 1 , ∞ , 1 + i, 0) S ( i ) = i- 1- (1 + i ) i- 1- · ∞- ∞- (1 + i ) =- 2 i- 1 = 1 + i 8. If T ( z ) = az + b cz + d show that T ( R ∞ ) = R ∞ iff we can choose a, b, c, d to be real numbers. ( → T ( R ∞ ) = R ∞ ) Let a = d and b = c = 0. Then T is the identity function which is an automorphism of the extended real numbers. Hence we have chosen the desired a, b, c, d ∈ R ( ← a, b, c, d ∈ R ) Since a, b, c, d are real and T acts on the extended real numbers, we note that T is a real function. So if suffices to show thatis a real function....
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This note was uploaded on 10/19/2009 for the course MATH 814 taught by Professor Cong during the Three '09 term at University of Adelaide.

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Home_7 - Homework 7 Stephen Taylor June 4, 2005 Pages 54-...

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