Lösungen

Aufgabe 1

Remove["Global`*"]

a

z1 = 2 + 4  I;  z2 = -4 - 6  I;

1/z1

N[%]

1/z2

N[%]

Conjugate[z1]

Re[z1]

b

Solve[z1 z  +  1/z2 == Conjugate[z1]/Re[z1], {z}]

Flatten[Solve[z1 z  +  1/z2 == Conjugate[z1]/Re[z1], {z}], 
      2]

zRes = 
      z /. Flatten[Solve[z1 z  +  1/z2 == Conjugate[z1]/Re[z1], {z}], 
          2];  zRes

N[%]

a, b

p[z_] := Point[{Re[z], Im[z]}]

Show[
        Graphics[
{PointSize[0.03], 
p[
              z1], p[z2], p[z2^(-1)], p[Conjugate[z1]], p[Re[z1]], 
            p[zRes], 
Line[{{0, -7}, {0, 7}}], 
            Line[{{-7, 0}, {7, 0}}], 
Circle[{0, 0}, 
              1]
}], AspectRatio -> 01];

Show[
        Graphics[
{PointSize[0.03], 
p[
              z2^(-1)], 
            p[zRes], 
Line[{{0, -1.1}, {0, 1.1}}], 
            Line[{{-1.1, 0}, {1.1, 0}}], 
Circle[{0, 
                0}, 1]
}], 
        AspectRatio -> 01];

c

Abs[z1]/ Abs[z2]

Abs[z1/z2]

N[%]

d, e

Solve[w^6 == z2, {w}]

solv1 = Solve[w^6 == z2, {w}] // N // Flatten

w[n_] := (w /. solv1[[n]])

w[4]

Show[
        Graphics[
Join[{PointSize[
                0.03], 
Line[{{0, -1.1}, {0, 1.1}}], 
              Line[{{-1.1, 0}, {1.1, 0}}], 
Circle[{0, 
                  0}, 1]
}, Table[p[w[n]], {n, 6}]]], 
        AspectRatio -> 01];

Show[
        Graphics[
Join[{PointSize[0.03], 
              p[z2], 
Line[{{0, -7}, {0, 7}}], 
              Line[{{-7, 0}, {7, 0}}], 
Circle[{0, 0}, 
                1], p[z2]
}, Table[p[w[n]], {n, 6}]]], 
        AspectRatio -> 1];

Aufgabe 2

Remove["Global`*"]

a, b

zA = -2 + I;  zB = 4 - I;  zC = 2 + 6  I;

φ = 5/11 Pi

zA1 =  zA E^(I φ)

N[%]

zB1 =  zB E^(I φ)

N[%]

zC1 =  zC E^(I φ)

N[%]

p[z_] := Point[{Re[z], Im[z]}];
q[z_] := {Re[z], Im[z]};
Show[
        Graphics[
{PointSize[0.03], 
p[
              zA], p[zA1], p[zB], p[zB1], p[zC], 
            p[zC1], 
Line[{q[zA], q[zB]}], 
            Line[{q[zA], q[zC]}], 
            Line[{q[zC], q[zB]}], 
Line[{q[zA1], q[zB1]}], 
            Line[{q[zA1], q[zC1]}], 
            Line[{q[zC1], 
                q[zB1]}], 
Line[{{0, -5}, {0, 5}}], 
            Line[{{-5, 0}, {5, 0}}]
}], 
        AspectRatio -> 01];

Aufgabe 3

f[x_] := 
      2 (x - 2) (x + 3) (x - 4) (x + 5) (x - 
            8)

a

Plot[f[x], {x, -6, 9}];

b

Solve[f[x] == 0, {x}]

c

5 NS

Remove["Global`*"]

Aufgabe 4

Cos[6 φ] // TrigExpand

Aufgabe 5

Remove["Global`*"]

<<Calculus`FourierTransform`

FourierTrigSeries[Cos[φ 2  Pi]^6, φ, 
      10]

FourierTrigSeries[Cos[φ 2  Pi]^6, φ, 10] // 
      Simplify

Aufgabe 6

Apart[(3 x^2 - 2  x + 1)/(x^2 + 2  x + 1)]

Aufgabe 7

Apart[1/((x - 2)^2 (x^2 + 2  x + 3))]

Aufgabe 8

Apart[1/((x - 2) (x + 3))]

Aufgabe 9

t = Solve[w^7 == 2 + 5  I, {w}] // Flatten

tab = Table[w[k] = w /. t[[k]], {k, 1, 7}]

Sum[tab[[k]], {k, 1, 7}] // N // Chop

Sum[tab[[k]], {k, 1, 7}] // ComplexExpand // 
      FullSimplify

Remove["Global`*"]

Aufgabe 10

Apart[x^3/((x - 1) (x + 2) (x + 4))]

Apart[x^2/((x - 1)^2 (x + 2))]

Apart[x^2/((x - 1)^4)]

Apart[1/(x (x - 1))]