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hw3sol

# hw3sol - Complex Analysis Spring 2001 Homework III...

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Unformatted text preview: Complex Analysis Spring 2001 Homework III Solutions 1. Conway, chapter 3, section 3, problem 8 If Tz = az + b cz + d show that T ( R ∞ ) = R ∞ if and only if a, b, c, d can be chosen to be real numbers. It is clear that if a, b, c, d are real, then T maps the extended real axis to the extended real axis. The qualification “if a, b, c, d can be chosen to be real numbers” is used, since the transformation T is unchanged if each of the coefficients is multiplied by a non-zero complex number. To prove the converse, we suppose that T maps the extended real axis to the extended real axis. We know that not both of c and d can be zero. We first investigate the case where c = 0 . Then T reduces to Tz = a d z + b d . Since the image of 0 is real, b d is real, and so is T 1- T 0 = a d . Together, these say that the coefficients can be taken to be real. If d = 0, then T ∞ = a c and T 1- T ∞ = b c are real, so the coefficients may be taken real. Finally, we consider the main case when neither c nor d is zero. Then T 0 = b d , T ∞ = a c , and T- 1 ∞ =- d c are all real. Taking the ratio of the first and third of these, b c is also real, and thus a, b, d are real multiples of c , so we choose a, b, c, d to be real. A better way to approach this problem was used by several people in their solutions. Since a M¨ obius transformation is determined by its action on three distinct points in the extended plane. Let x 1 , x 2 , and x 3 be distinct real numbers whose images under T , say r 1 , r 2 , and r 3 are distinct finite real numbers also. Then by preservation of cross ratio, ( Tz, r 1 , r 2 , r 3 ) = ( z, x 1 , x 2 , x 3 ) . Writing this out explicitly with w = Tz , w- r 2 w- r 3 · r 1- r 3 r 1- r 2 = z- x 2 z- x 3 · x 1- x 3 x 1- x 2 , which can be written as w- r 2 w- r 3 = β z- x 2 z- x 3 where β is real. (It is the ratio of non-zero real ratios.) But now each side is a M¨ obius transformation with real coefficients, and the inverse of the transformation on the left...
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hw3sol - Complex Analysis Spring 2001 Homework III...

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