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HW 2 for Math 419, Fall 2009 Due Friday, September 4 8/5d: By writing the individual factors on the left in exponential form, performing the needed operations, and finally changing back to rectangular √ √ −10 −11 = 2 (−1 + 3i). coordinates, show that (1 + 3i) 8/10a: Use de Moivre’s formula to derive the trig identity cos 3θ = cos3 θ − 3 cos θ sin2 θ. 10/6:√ Find the four zeros of the polynomial z 4 + 4, one of them being z0 = 2eiπ/4 = 1 + i. Then use those zeros to factor z 4 + 4 into quadratic factors with real coefficients. (Hint: The answer is (z 2 +2z +2)(z 2 −2z +2).) 10/7: Show that if c is any nth root of unity other than unity itself, then 1 + c + c2 + . . . + cn−1 = 0. (Hint: Use the first part of 8/9 from hw 1.) 11/1: Sketch the following sets and determine which are domains: (a) |z − 2 + i| ≤ 1; (b) |2z + 3| > 4; (c) Im(z) > 1; (d) Im(z) = 1; π (e) 0 ≤ arg(z) ≤ 4 (z 6= 0); (f) |z − 4| ≥ |z|. (Hint: (b) and (c) are domains.) 11/2: Which sets in Exercise 1 (the previous one) are neither open nor closed? (Hint: (e).) 11/3: Which sets in Exercise 1 are bounded? (Hint: (a).) 11/9: Show that any point z0 of a domain is an accumulation point of that domain. 11/10: Prove that a finite set of points z1 , z2 , . . . , zn cannot have any accumulation points. 12/1: For each of the functions below, describe the domain of definition that is understood: (b) f (z) = Arg( 1z ); (a) f (z) = z 21+1 ; z 1 (c) f (z) = z+z ; (d) f (z) = 1−|z| 2. (Hint: (a) z 6= ±i, (c) Re(z) 6= 0.) 12/3: Suppose that f (z) = x2 − y 2 − 2y + i(2x − 2xy), where z = x + iy. Use the expressions z+z z−z x= and y = 2 2i to write f (z) in terms of z, and simplify the result. (Hint: f (z) = z 2 + 2iz.)