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# 01 - ASSIGNMENT 1 Â SOLUTIONS MAT 572 A Â FALL 2007 Problem...

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Unformatted text preview: ASSIGNMENT 1 Â· SOLUTIONS MAT 572 A Â· FALL 2007 Problem 1 ( cf. [1, Exercise I.4.5]) . Let z = cis(2 Ï€/n ) for an integer n â‰¥ 2. Show that 1 + z + z 2 + Â·Â·Â· + z n- 1 = 0 . Proof. Let s denote the value of the sum. Since z n = 1, we have zs = z + z 2 + Â·Â·Â· + z n- 1 + z n = z + z 2 + Â·Â·Â· + z n- 1 + 1 = s. Thus (1- z ) s = s- zs = 0. Since n â‰¥ 2, we have z 6 = 1, and it follows that s = 0. Incidentally, for a geometric interpretation of this trick, note that the set { 1 ,z,z 2 ,...,z n- 1 } of n points evenly distributed around the unit circle is invariant under the rigid motion of rotation by 2 Ï€/n radians; that is, under the transformation w 7â†’ zw . Therefore the sum over all the elements in the set must also be invariant. But the only complex number invariant under a non-trivial rotation is 0. Problem 2 ( cf. [1, Exercise II.1.11]) . (a) Show that the set S = { cis k | k âˆˆ N } is dense in T = { z âˆˆ C | | z | = 1 } ....
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01 - ASSIGNMENT 1 Â SOLUTIONS MAT 572 A Â FALL 2007 Problem...

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