hw1 - of the (extended) complex plane to itself: inversion,...

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Complex Analysis Spring 2001 Homework I Due Friday April 13 1. Conway, Chapter 1, section 3, problem 3. 2. The equation az + b ¯ z + c = 0 can have solution set consisting of a single point, or a line in the complex plane. (a) Find necessary and sufficient conditions on the complex constants a, b , and c so that the solution set is a line. (b) Find the solution in the case it is a single point. 3. The next two questions pertain to the correspondence between the extended complex plane and the unit sphere in R 3 given in Chapter 1, section 6. (a) Find the mapping of the sphere to itself induced by the following two mappings
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Unformatted text preview: of the (extended) complex plane to itself: inversion, in which z 1 z for z 6 = 0, 0 and 0 ; and inversion in the unit circle, where z 1 z for z 6 = 0, with 0 and 0 . (b) Show that non-zero complex numbers z and z correspond to diametrically op-posed points on the sphere if and only if z z =-1 . 4. Conway, chapter 3, section 1, problem 7. 5. Conway, chapter 3, section 2, problem 9. 6. Conway, chapter 3, section 2, problem 14....
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