Die Joukowski-Funktion

Needs["Graphics`ComplexMap`"]

\(plot1 = 
      PolarMap[\(1\/2\) \((# + 1\/#)\) &, {0.1, 1}, {0, 2  π}, 
        PlotRange -> {\(-3\), 3}]\)

Das Joukowski-Profil

Clear[Realteil, Imaginärteil]

    
Realteil[f_] := 
      Simplify[ComplexExpand[Re[f /. z -> x + I y], 
          TargetFunctions -> Conjugate]]

    
Imaginärteil[f_] := 
      Simplify[ComplexExpand[Im[f /. z -> x + I y], 
          TargetFunctions -> Conjugate]]

u = Realteil[(1/2) (z + (1/z))]

v = Imaginärteil[(1/2) (z + (1/z))]

x = -0.2 + 1.3  Cos[t];
y = 0.5 + 1.3  Sin[t];

plot2 = ParametricPlot[{u, v}, {t, 0, 2  π}]

Clear[x, y]

Umkehrfunktion

sol = Solve[w == (1/2) (z + (1/z)), z]

plot1 = 
      CartesianMap[z /. sol[[1]] /. w -> # &, {-2, 
          2}, {-1, 1}]

plot2 = 
      CartesianMap[z /. sol[[2]] /. w -> # &, {-2, 
          2}, {-1, 1}]

Show[plot1, plot2]

Kurvenintegrale

(∫_K^□)(((z - 
                z0))^m) dz

int = (∫_0^(2  π))((((z[t] - z0))^m) z'[
                  t] /. {z -> (z0 + 
                        r (E^(I  #)) &)}) \[DifferentialD]t

Simplify[int, m ∈ Integers]

int = (∫_0^(2  π))((((z[t] - z0))^(-1)) z'[
                  t] /. {z -> (z0 + 
                        r (E^(I  #)) &)}) \[DifferentialD]t

\(TraditionalForm\`∫\_Q\%□\ \(z\^2\) dz\)

(Subscript[Γ, 1])

(∫_0^1)(z[t]^2 z'[
                t] /. {z -> (# &)}) \[DifferentialD]t

(Subscript[Γ, 2])

(∫_0^1)(z[t]^2 z'[
                t] /. {z -> (1 + #  I &)}) \[DifferentialD]t

Summe

int1 = (∫_0^1)(z[t]^2 z'[
                    t] /. {z -> (# &)}) \[DifferentialD]t + (∫_0^1)(z[t]^2 z'[
                    t] /. {z -> (1 + #  I &)}) \[DifferentialD]t

(Subscript[Γ, 3])

(∫_0^1)(z[t]^2 z'[
                t] /. {z -> (#  I &)}) \[DifferentialD]t

(Subscript[Γ, 4])

(∫_0^1)(z[t]^2 z'[
                t] /. {z -> (# + I &)}) \[DifferentialD]t

Summe

int2 = (∫_0^1)(z[t]^2 z'[
                    t] /. {z -> (#  I &)}) \[DifferentialD]t + (∫_0^1)(z[t]^2 z'[
                    t] /. {z -> (# + I &)}) \[DifferentialD]t

int1 - int2

(∫_Q^□) Conjugate[
              z]^2 dz

int1 = (∫_0^1)(Conjugate[z[t]]^2 z'[
                    t] /. {z -> (# &)}) \[DifferentialD]t + (∫_0^1)(Conjugate[z[t]]^2 z'[
                    t] /. {z -> (1 + #  I &)}) \[DifferentialD]t

int1 = (int1 /. {Conjugate[t] -> t})

int2 = (∫_0^1)(Conjugate[z[t]]^2 z'[
                    t] /. {z -> (#  I &)}) \[DifferentialD]t + (∫_0^1)(Conjugate[z[t]]^2 z'[
                    t] /. {z -> (# + I &)}) \[DifferentialD]t

int2 = int2 /. {Conjugate[t] -> t}

int1 - int2