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4 - unity(g Since the exponent is greater than one there is...

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2 I.2 1. Note that the problem asks for all solutions. (c) If z = 4 - 1, then z 4 = - 1, thus z 8 = 1, so we only need consider 8th roots of unity. Since z 4 ± = 1, this rules out 1 , - 1 , i, - i .A simple calculation reveals that the other four are solutions, thus the solutions are e i π / 4 ,e 3 i π / 4 ,e 5 π / 4 ,e 7 π / 4 . In Cartesian coordinates, these are 2+ i 2 2 , - 2+ i 2 2 , - 2 - i 2 2 , 2 - i 2 2 . (f) If z = (3 - 4 i ) 1 / 8 , then z 8 =3 - 4 i . If we let θ = arctan - 4 / 3, then 3 - 4 i =5 e i θ . Thus 5 1 / 8 e i θ / 8 is one solution. To Fnd all eight solutions, just multiply this one by each of the 8th roots of
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Unformatted text preview: unity. (g) Since the exponent is greater than one, there is only one solution. To Fnd it, Frst note that (1+ i ) 2 = 2 i . Thus (1+ i ) 8 = (2 i ) 4 = 16. 5. (a) Let S = n ± j =0 z j . Then we have zS-S = n +1 ± j =1 z j-n ± j =0 z j = z n +1-1 . Dividing both sides by 1-z gives S = 1-z n +1 1-z (b) Noticing that cos j θ = ² ² e j θ ³ , 4...
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