The Complex Exponential (Subscript[E, λ])(
          z) = λ (e^z)

This notebook helps you to visualize some of the properties of the complex exponential function. The escape criterion used in this program is the one explained in Chapter 18 of the book.

Filled Julia sets for (Subscript[E, λ])

The first cell compiles the function and the second one draws the picture. Choose a value of λ to see the corresponding filled Julia set for this family.

Iterates = 40;
\(\(Julia = 
        Compile[{{z, _Complex}, {λ, _Complex}, {Iterates, _Integer}}, 
          Length[FixedPointList[λ\ E\^# &, z, Iterates, 
              SameTest -> \((Re[#2] > 
                      50.  &)\)]]];\)\)

λ = (1/E);

    
RealZInf = 1;
RealZSup = 8;
ImagZInf = -2;
ImagZSup = 
        2;


$TextStyle = {FontFamily -> "Times", 
          FontSize -> 14};
FilledJuliaSet = 
        DensityPlot[
          Julia[x + I y, λ, Iterates], {x, RealZInf, RealZSup}, {y, 
            ImagZInf, ImagZSup}, Mesh -> False,  Frame -> True, 
          Axes -> False, PlotPoints -> 200, 
          AspectRatio -> Automatic, 
          FrameLabel -> {"Re(z)", "Im(z)"}, 
          RotateLabel -> False, 
          PlotLabel -> "Filled Julia set for λ =" <> 
              ToString[N[λ]], 
          ColorFunction -> (If[# ≥ 1, RGBColor[0, 0, 0], 
                  Hue[#]] &), 
          ImageSize -> 600];
Clear[λ];

Parameter λ-Plane

This cell plots the parameter plane for λ. Choose which portion of the set you want to display by changing the corners of the window.

Iterates = 40;
\(\(λPlane = 
        Compile[{{λ, _Complex}, {Iterates, _Integer}}, 
          Length[FixedPointList[λ\ E\^#\  &, λ, Iterates, 
              SameTest -> \((Re[#2] > 
                      50. \  &)\)]]];\)\)

λrealInf = -2;
λrealSup = 6;
λimagInf = -4;
λimagSup = 4;

    
\($TextStyle = {FontFamily -> "Times", FontSize -> 14}; 
    ParameterPlane = 
      DensityPlot[λPlane[x + I\ y, 
          Iterates], {x, λrealInf, λrealSup}, {y, λimagInf, λimagSup}, Mesh -> False, \ 
        Frame -> True, Axes -> False, PlotPoints -> 200, 
        AspectRatio -> Automatic, 
        FrameLabel -> {"Re(λ)", "Im(λ)"}, 
        PlotLabel -> "The λ-plane", RotateLabel -> False, 
        ColorFunction -> \((If[# ≥ 1, RGBColor[0, 0, 0], 
                Hue[#]] &)\), ImageSize -> 500];\)