Download November 15, 2016 Practice (Analyse 3 NA) Exercises 33–36 33

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