Lösungen

1

m1 = Solve[(z - 2 + I)^4 == 2 + I, {z}] // Flatten // 
        N

f[x_] := x[[2]];     f[Rule[c, b]]

m2 = Map[f, m1]

m3 = Map[Abs, m2]

2

3  E^(2  z) /. z -> I Pi

(3  E^(2  z) /. z -> I Pi + 1)  //  Simplify

(Cos[z] + Sin[z] /. z -> I Pi ) // N

(Cos[z] - Sin[z] /. z -> -I Pi ) // N

3

z = 4 + 2  I

w = 1/z

Show[
        Graphics[{PointSize[0.03], Point[{Re[z], Im[z]}], 
            Point[{Re[w], Im[w]}], Point[{0, 0}], Circle[{0, 0}, 1],  
            Line[{{-1.2, 0}, {4, 0}}],  Line[{{0, -1.2}, {0, 2}}], 
            Line[{{0, 0}, {Re[z], Im[z]}}], 
Line[{{0, 0}, 
                10 {Re[w], Im[w]}}]}], 
        AspectRatio -> Automatic];

(4 + 2  I) (2 - 4  I)/(4 - 3  I)

(4 + 2  I) (2 - 4  I)/(3 + 4  I)

4

Remove[z, w]

z1 = 1; z2 = -1; z3 = 2 - I; z4 = 2 + I; z5 = -2 + I; 
    z6 = -2 - I;
(z - z1) (z - z2) (z - z3) (z - z4) (z - z5) (z - 
            z6) // Expand

((z - z1) (z - z2) (z - z3) (z - z4) (z - 
                z5) (z - z6) // Expand)  /. 
      z -> w^(1/2)

Solve[% == 0, {w}]

5

a = (z^12 - 1)/(2  z - 2)  // Apart

a /. z -> 1

a /. z -> I

a /. z -> -1

a /. z -> -I