Q  9.2  fz=(Q-I Γ)/(2 Pi)Log[z]    p.231

fz = (Q - I Γ)/(2 Pi) Log[z]

V = Exp[I Im[Log[Conjugate[D[fz, z]] /. z -> z[t]]]]

step1  Q=0  Γ=1

eq = z'[t] == 
          V /. {Q ->  0, Γ ->  1}

ne = 5;
z0 = 3 E Table[Exp[-i], {i, ne}]

zuall = 
      Table[{ans = NDSolve[{eq, z[0] == z0[[i]]}, 
                z[t], {t, 0, 25}][[1]]; 
          ParametricPlot[Evaluate[{Re[z[t]], Im[z[t]]} /. ans], {t, 0, 25}, 
            PlotStyle ->  Hue[0.7], 
            PlotRange ->  {{-3, 3}, {-3, 3}}, 
            DisplayFunction ->  Identity]}, {i, ne}]

zu1 = 
      Show[zuall, DisplayFunction ->  $DisplayFunction, 
        AspectRatio ->  Automatic]

step2  Q=1  Γ=0

eq = z'[t] == 
          V /. {Q ->  1, Γ ->  0}

ne = 12
z0 = 
      Table[0.01 Exp[I 360*i/ne °], {i, 
          ne}]

zuall = 
      Table[{ans = NDSolve[{eq, z[0] == z0[[i]]}, 
                z[t], {t, 0, 20}][[1]]; 
          ParametricPlot[Evaluate[{Re[z[t]], Im[z[t]]} /. ans], {t, 0, 20}, 
            PlotStyle ->  Hue[0.7], 
            PlotRange ->  {{-3, 3}, {-3, 3}}, 
            DisplayFunction ->  Identity]}, {i, ne}]

zu2 = 
      Show[zuall, DisplayFunction ->  $DisplayFunction, 
        AspectRatio ->  Automatic]

step3  Q=1  Γ=1

eq = z'[t] == 
          V /. {Q ->  1, Γ ->  1}

ne = 12
z0 = Table[0.01 Exp[I 360*i/ne °], {i, ne}]

zuall = 
      Table[{ans = NDSolve[{eq, z[0] == z0[[i]]}, 
                z[t], {t, 0, 20}][[1]]; 
          ParametricPlot[Evaluate[{Re[z[t]], Im[z[t]]} /. ans], {t, 0, 20}, 
            PlotStyle ->  Hue[0.7], 
            PlotRange ->  {{-3, 3}, {-3, 3}}, 
            DisplayFunction ->  Identity]}, {i, ne}]

zu3 = 
      Show[zuall, DisplayFunction ->  $DisplayFunction, 
        AspectRatio ->  Automatic]

summary

Show[GraphicsArray[{{zu1, zu3, zu2}}]]