Argp[z_] :=  
      If[θ < 0, θ + 2 Pi, θ] /. θ ->  
          Im[Log[z]]

Argp[z]

Argp[1.  + I]/Degree
Argp[1.  - I]/Degree

Q 9.1   fz=z^n; f'(z)=n z^(n-1)  p. 230

eq = z'[
          t]==Exp[-I (n - 1) Argp[
              z[t]]]

step1  n=1

ne = 10
z0 = -30 + 
          0.5 Exp[I 90  °] Table[i, {i, ne}];

zuall = 
      Table[{ans = NDSolve[{eq /. n ->  1, 
                  z[0] == z0[[i]]}, z[t], {t, 0, 200}][[1]]; 
          ParametricPlot[Evaluate[{Re[z[t]], Im[z[t]]} /. ans], {t, 0, 200}, 
            PlotStyle ->  Hue[0.7], 
            PlotRange ->  {{-5, 5}, {0, 3}}, 
            DisplayFunction ->  Identity]}, {i, ne}]

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

step2  n=2

ne0 = 10;
z01 = 
        I 30 + 0.04 Exp[I 0  °] 
            Table[i, {i, ne0}];
z02 = 
        I 30 - 0.04 Exp[I 0  °] 
            Table[i, {i, ne0}];
z0 = Join[z01, z02];
ne = Length[z0];

zuall = 
      Table[{ans = NDSolve[{eq /. n ->  2, 
                  z[0] == z0[[i]]}, z[t], {t, 0, 100}][[1]]; 
          ParametricPlot[Evaluate[{Re[z[t]], Im[z[t]]} /. ans], {t, 0, 100}, 
            PlotStyle ->  Hue[0.7], 
            PlotRange ->  {{-5, 5}, {0, 3}}, 
            DisplayFunction ->  Identity]}, {i, ne}]

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

step3  n=3

ne0 = 5;
z01 = 
        30 Exp[I  60 °] - 
          0.0025 Exp[I 150  °] 
            Table[i, {i, ne0}];
z02 = 
        30 Exp[I  60 °] + 
          0.0025 Exp[I 150  °] 
            Table[i, {i, ne0}];
z03 = 
        30 Exp[I  180 °] + 
          0.0025 Exp[I 90  °] 
            Table[i, {i, ne0}];
z0 = Join[z01, z02, z03];
ne = Length[z0];

zuall3 = 
      Table[{ans = NDSolve[{eq /. n ->  3, 
                  z[0] == z0[[i]]}, z[t], {t, 0, 100}][[1]]; 
          ParametricPlot[Evaluate[{Re[z[t]], Im[z[t]]} /. ans], {t, 0, 100}, 
            PlotStyle ->  Hue[0.7], 
            PlotRange ->  {{-50, 50}, {0, 10}}, 
            DisplayFunction ->  Identity]}, {i, ne}]

zu3 = 
      Show[zuall3, PlotRange ->  {{-5, 5}, {0, 5}}, 
        DisplayFunction ->  $DisplayFunction, 
        AspectRatio ->  Automatic]

step4  n=4

ne0 = 10;
z01 = 
        30 Exp[I 45  °] + 
          0.001 Exp[I 135  °] 
            Table[i, {i, ne0}];
z02 = 
        30 Exp[I 45  °] - 
          0.001 Exp[I 135  °] 
            Table[i, {i, ne0}];
z03 = 
        30 Exp[I 135  °] + 
          0.001 Exp[I 45  °] 
            Table[i, {i, ne0}];
z04 = 
        30 Exp[I 135  °] - 
          0.001 Exp[I 45  °] 
            Table[i, {i, ne0}];
z0 = Join[z01, z02, z03, z04];
ne = Length[z0];

zuall = 
      Table[{ans = NDSolve[{eq /. n ->  4, 
                  z[0] == z0[[i]]}, z[t], {t, 0, 100}][[1]]; 
          ParametricPlot[Evaluate[{Re[z[t]], Im[z[t]]} /. ans], {t, 0, 100}, 
            PlotStyle ->  Hue[0.7], 
            PlotRange ->  {{-5, 5}, {0, 5}}, 
            DisplayFunction ->  Identity]}, {i, ne}]

zu4 = 
      Show[zuall, DisplayFunction ->  $DisplayFunction, 
        AspectRatio ->  Automatic]

step5  n=3/4

ne = 12
z0 = 
        30 Exp[-120 ° I ] + 
          0.75 Exp[I 150  °] Table[i, {i, ne}];

zuall = 
      Table[{ans = NDSolve[{eq /. n ->  3/4, 
                  z[0] == z0[[i]]}, z[t], {t, 0, 100}][[1]]; 
          ParametricPlot[Evaluate[{Re[z[t]], Im[z[t]]} /. ans], {t, 0, 100}, 
            PlotStyle ->  Hue[0.7], 
            PlotRange ->  {{-5, 5}, {-3, 3}}, 
            DisplayFunction ->  Identity]}, {i, ne}]

zu5 = 
      Show[zuall, DisplayFunction ->  $DisplayFunction, 
        AspectRatio ->  Automatic]

step6  n=2/3

ne = 20
z0 = 
        30 Exp[-90 ° I ] + 
          0.75 Exp[I 180  °] Table[i, {i, ne}];

zuall = 
      Table[{ans = NDSolve[{eq /. n ->  2/3, 
                  z[0] == z0[[i]]}, z[t], {t, 0, 200}][[1]]; 
          ParametricPlot[Evaluate[{Re[z[t]], Im[z[t]]} /. ans], {t, 0, 200}, 
            PlotStyle ->  Hue[0.7], 
            PlotRange ->  {{-5, 5}, {-3, 3}}, 
            DisplayFunction ->  Identity]}, {i, ne}]

zu6 = 
      Show[zuall, DisplayFunction ->  $DisplayFunction, 
        AspectRatio ->  Automatic]

step7  n=1/2

ne = 20
z0 = 30 + 1 Exp[I 270  °] Table[i, {i, ne}];

zuall = 
      Table[{ans = NDSolve[{eq /. n ->  1/2, 
                  z[0] == z0[[i]]}, z[t], {t, 0, 500}][[1]]; 
          ParametricPlot[Evaluate[{Re[z[t]], Im[z[t]]} /. ans], {t, 0, 500}, 
            PlotStyle ->  Hue[0.7], 
            PlotRange ->  {{-2, 4}, {-3, 3}}, 
            PlotPoints ->  500, 
            DisplayFunction ->  Identity]}, {i, ne}]

zu7 = 
      Show[zuall, DisplayFunction ->  $DisplayFunction, 
        AspectRatio ->  Automatic]

step8  n=3/2   New

ne = 25;
z0 = 
        30 Exp[I 120  ° ]  + 
          0.075 Exp[I 30  °] Table[i, {i, ne}];

zuall = 
      Table[{ans = NDSolve[{eq /. n ->  3/2, 
                  z[0] == z0[[i]]}, z[t], {t, 0, 100}][[1]]; 
          ParametricPlot[Evaluate[{Re[z[t]], Im[z[t]]} /. ans], {t, 0, 100}, 
            PlotStyle ->  Hue[0.7], 
            PlotRange ->  {{-5, 5}, {0, 3}}, 
            DisplayFunction ->  Identity]}, {i, ne}]

zu8 = 
      Show[zuall, DisplayFunction ->  $DisplayFunction, 
        AspectRatio ->  Automatic]

step81  n=3/2   New and better constant speed of stream line  for check

fz81 = z^(3/2);
eq81 = z'[t]==Conjugate[
          D[fz81, z] /. z -> z[t]]

zuall81 = 
      Table[{ans = NDSolve[{eq81, z[0] == z0[[i]]}, 
                z[t], {t, 0, 100}][[1]]; 
          ParametricPlot[Evaluate[{Re[z[t]], Im[z[t]]} /. ans], {t, 0, 100}, 
            PlotStyle ->  Hue[0.9], 
            PlotRange ->  {{-5, 5}, {0, 3}}, 
            DisplayFunction ->  Identity]}, {i, ne}]

zu81 = 
      Show[zuall81, DisplayFunction ->  $DisplayFunction, 
        AspectRatio ->  Automatic]

Show[zu8, 
      zu81]

summary

Show[
      GraphicsArray[{{zu5, zu6, zu7}, {zu1, zu2, zu3}, {zu4, 
            zu8}}]]