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November 15, 2016 Practice (Analyse 3 NA) Exercises 33–36 33. (Manipulating complex numbers) (a) Given are the complex numbers z = 3 + 4i and w = 2 − i, compute z/w. (b) Given v = 4 − 4i, determine |v|, arg v and write v in polar coordinates v = r eiφ . (c) Given the information that a complex number w fulfills |w| = 8 and arg w = 5π/6, compute Re(w) and Im(w). (d) Give a geometric construction of the complex numbers computed in (a) using only a ruler and a protractor. (e) Demonstrate that z + w = z + w and that z w = z w. (f) Let c > 0 and c 6= 1. Show that the complex numbers z, for which the equality z − i z + i = c holds true, form a circle. (Hint: Write z = x + iy to get an equation for x and y.) 34. (de Moivre’s formula) (a) Write cos(3φ) and sin(3φ) in terms of cos(φ) and sin(φ). √ (b) Write ( 3 + i)10 in the form a + ib. 35. (Finding roots, factorization of polynomials and multiplicity of roots) (a) Compute the roots of the polynomial and write them in the form a + ib. (i) z 2 + iz + 2 (ii) z 2 + (6i − 8) (iii) z 3 + 64 (give the answer also in polar coordinates) (iv) z 4 + 4 (give the answer also in polar coordinates) (b) (i) Decompose z 4 + 4 into linear factors. (ii) The polynomial z 4 + 4 cannot be factored linearly in the reals. Find a factorization of the polynomial into two factors of order 2 in the reals. (iii) Find the roots of z 5 − 3z 4 − z 3 + 7z 2 − 4 and determine the multiplicity of the root z = 2. 36. (Elementary functions) Write the following expressions in the form a + ib: √ √ ln(−i), ln(i) + ln(−1) , sin−1 (2) , tan−1 (i) , ( 2 + 2i)(i+1) . Solutions: 33. d) Constructing z/w without any computation, but simply by measuring lengths and angles: Figure 1: The angle θ1 − θ2 can be constructed with a protractor, the length of through equal ratios in similar triangles r1 r2 is constructed