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Unformatted text preview: 18.03 Problem Set 3 Spring 2009 Due in boxes at Room 2-106, 12:55 pm, Friday, Feb 20 Part A (16 points) (at due date) Write the names of any person, web site, or materials you consulted. Write “No C” (no consultation) on your paper if you consulted no fellow students, outside mate- rials/people. Lec 5, Fri, Feb 13; Complex numbers; complex exponentials. Read: C.1-3, Work: 2E-1, 2, 7, 14, 16b Lec 6, Tue, Feb 17; More about exponentials; Roots of unity Read: C.4, IR.6 Work: 2E-9,10 Lec 7, Wed, Feb 18; Linear response to exponential and sinusoidal input. Read: IR.1-3, 5; Work: 2E-15 Lec 8, Fri, Feb 20; Autonomous equations, phase line, stability. Read: EP 1.7, 7.1 Part B (32 points) 1. (Lec 5, Fri, Feb 13) [9pts: 2,2,1,2,2] Complex numbers; polar representation a) Express 2 / (1 + i ) as a + bi and as re iθ where a , b , r > 0, and θ are real numbers. b) Find the real an imaginary parts of e 2- πi and e 2+ πi/ 2 . c) Suppose that z 3 = 1 and z 6 = 1. Show that z 2 + z + 1 = 0 using algebra. d) Do part (c) again by a more circuitous method. i) Find the polar representations for the two root of z 3 = 1 with z 6 = 1, and use this to find the rectangular representations (the form...
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This note was uploaded on 05/06/2010 for the course 18 18.03 taught by Professor Unknown during the Spring '09 term at MIT.
- Spring '09