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Unformatted text preview: • On occasion, a problem set contains one or more problems designated as “op- tional.” We do NOT grade such problems. Nevertheless, you are responsible for learning the subject matter within their scope. Overview This problem set is designed to strengthen your mathematical foundations. In par- ticular, you will get practice with complex numbers and their arithmetic and complex variables and their algebra; you will also tiptoe into the realm of complex-valued func- tions (with more to come later). This problem set also explores some of the salient properties of the Dirac delta. Getting comfortable with the geometry of the complex plane and developing fluency with using a graphical approach to solve problems that involve complex numbers, variables, and functions should be among your main achievements this week. The scope of this problem set includes subject matter covered in lectures, discussion sections, and office hours up to, and including, 16 September 2010. Reading Please read the following sections of the textbook (Lee & Varaiya), in the following order: • Appendix B • Chapter 1. • Chapter 2. You should also visit the “Shortcuts” sidebar on the Ptolemy Web site, where you will find a link to a Java applet demonstrating the behavior of phasors. From the Ptolemy Web site, follow the sidebar sequence Shortcuts → To Applets → Phasors. Alternatively, visit the following URL: http://ptolemy.eecs.berkeley.edu/eecs20/berkeley/phasors/demo/phasors.html . 2 HW1.1 Two complex numbers z 1 and z 2 are described below: z 1 = 1 + i √ 3 z 2 = exp parenleftbigg i 2 π 3 parenrightbigg . (a) Identify each of the following complex numbers as points (or vectors) on the complex plane, using a well-labeled sketch: z 1 , z 2 , z ∗ 1 , z ∗ 2 , 1 /z 1 , 1 /z 2 , 1 /z ∗ 1 , and 1 /z ∗ 2 . (b) Determine each of the sums z 1 + 2 z 2 , z 2 1 + z 2 , and 1 2 z 1 + z ∗ 2 . (c) Determine each of the magnitudes | z 1 z 2 | , | z 1 z ∗ 2 | , | z 1 /z 2 | , and | z 2 /z 1 | . (d) Determine each of the following powers of z 1 and z 2 : (i) z 2 1 (ii) z 3 1 (iii) z 6 1 (iv) z 4 2 . (e) Determine z 1 / 4 2 . Be mindful of how many fourth roots z 2 has and identify each of them graphically on a well-labeled sketch of the complex plane. Express each of your answers in Cartesian form ( a + ib ), in polar form ( re iθ , where r > 0), as a real number, as an imaginary number, or graphically in a well-labeled complex-plane diagram, whichever form is less cluttered and more appropriate. HW1.2 Exercise 13 of Appendix B of Lee & Varaiya, and the supplementary problem (d) provided below. (c) Hint : Recognize that A cos( ωt + φ ) = n summationdisplay k =1 A k cos( ωt + φ k ) can be obtained by equating the real parts of the two sides of Ae i ( ωt + φ ) = n summationdisplay k =1 A k e i ( ωt + φ k ) ....
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- Spring '09
- Complex number, Dirac delta function, Dirac delta