Complex numbers, complex exponentials 1. Mark z = 1 + √3i on the complex plane. What is its polar coordinates? Then mark zn for n = 1, 2, 3, 4. What is each in the form a + bi? What is each none in the form Aeiθ? Then mark z for n = 0, −1, −2, −3, −4. 2. Find a complex number a+bi such that ea+bi = 1+√3i. In fact, find all such complex numbers. For definiteness, fix b to be positive but as small as possible. (This is probably the first one you thought of.) What is en(a+bi) for n = 1, 2, 3, 4? (Hint: en(a+bi) = (ea+bi)n.) How about for n = 0, −1, −2, −3, −4? 3. Write each of the following functions f (t) in the form A cos(ωt−φ). In each case, begin by drawing a right triangle with sides a and b. (a) cos(2t)+sin(2t). e(b) cos(πt) −√3 sin(πt). (c) Re it . 2 + 2i 4. Find a solution of ˙x+2x = et of the form wet . Do the same for ˙z +2z = e2it . In both cases, go on to write down the general solution. 5. Find a solution of ˙x + 2x = cos(2t) by replacing it with a complex valued equation, solving that, and then extracting the real part. Your work also gives you a solution for ˙x + 2x = sin(2t). Recitation 5, February 18, 2010MIT OpenCourseWare http://ocw.mit.edu 18.03 Differential Equations���� Spring 2010 For information about citing these materials or our Terms of Use, visit:
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