extra_6 - Extra Problems Sheet 6 Stephen Taylor May 10,...

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Extra Problems Sheet 6 Stephen Taylor May 10, 2005 1. Show that a M¨ obius transformation T ( z ) can have at most two fixed points in the complex plane unless T ( z ) z . For a 0 , b 0 , c 0 , d 0 C , let T ( z ) a 0 z + b 0 c 0 z + d 0 be a M¨ obius transformation. Since we may assume a 0 6 = 0 we multiply T ( z ) by a - 1 0 /a - 1 0 to obtain T ( z ) = z + b 0 /a 0 c 0 /a 0 z + d 0 /a 0 z + a bz + c from which we observe that the transformation has three degrees of freedom. This follows analytically from the fact that z = z + b cz + d z = (1 - d ) ± p ( d - 1) 2 - 4 cb 2 c are the fixed points of the transformation. ± 2. Discuss the image of the circle | z - 2 | = 1 and its interior under the map T ( z ) = z - 1 . Since this map is a M¨ obius transformation if maps circles to circles. So it suffices to check the image of three points on the given domain to obtain the image of the entire map. So we note 3 1 / 3 2 + i 1 / 5(2 - i ) 1 1 . So the image of the circle is the unit circle determined by the points
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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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extra_6 - Extra Problems Sheet 6 Stephen Taylor May 10,...

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