solutionshw7-201141

solutionshw7-201141 - Homework 7 1. Express in the form a +...

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Homework 7 1. Express in the form a + bi . a. (1 + i ) 20 . Solution: 1 + i = 2 e πi 4 . Hence (1 + i ) 20 = 2 10 e 5 πi = - 2 10 . b. 1 - 2 i 2+ i . Solution: 1 - 2 i 2+ i = 1 - 2 i 2+ i 2 - i 2 - i = - 5 i 5 = - i . 2. Solve z 2 - 4 z + (4 + 2 i ) = 0. Solution: z 2 - 4 z + (4 + 2 i ) = ( z - 2) 2 + 2 i = 0 gives ( z - 2) 2 = - 2 i = 2 e - πi 2 . Hence z - 2 = 2 e - πi 4 + kπi for k = 0 , 1. It follows that z = 2 + 2( 1 2 2 - 1 2 i 2) = 3 - i and z = 2 - 2( 1 2 2 - 1 2 i 2) = 1 + i . 3. Describe the sets whose points satisfy the following relations. Which of these sets are regions (i.e., open and connected sets)? a. | z + i | ≤ 1. Solution: | z + i | ≤ 1 ⇐⇒ the distance from z to - i is 1, i.e., the closed disk with center - i and radius 1. This set is not open. b. ± ± z - 1 z +1 ± ± = 1. Solution: ± ± z - 1 z +1 ± ± = 1 ⇐⇒ the distance of z to 1 equals the distance to - 1, i.e. the perpendicular bisector of the line segment connecting 1 and - 1 or the imaginary axis. This set is not open.
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solutionshw7-201141 - Homework 7 1. Express in the form a +...

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