Zhukovskii-Abbildung

Zhu[z]:=z + 1 / z

Wähle zuerst im Menü Kernel, Evaluation -> Evaluate Notebook !!

m[z_] := (z + 1)/(z - 1);

InvZhu1[z_] := m[Sqrt[m[0.5*z]]];

InvZhu2[z_] := m[-1*Sqrt[m[0.5*z]]];

\(\(InvZhu[z_]\  := \ \ 
      If[Sign[Im[InvZhu1[z]]] == Sign[Im[z]], 

        																								N[InvZhu1[z], 4], 

        																								N[InvZhu2[z], 4]

        																			\ \ \ \ \ ]; \)\)

Zhu[z_] := z  +  1/z ;

\(winkel\  := 30; 
k := \(-0.25\) + 0.25\ I; 
tr[z_] := z + k; 

    streckung\  := Sqrt[\((1 - Re[k])\)^2 + \((Im[k])\)^2]; 

    d[z_] := \((Cos[Pi\ winkel/180] + I\ Sin[Pi\ winkel/180])\)\ z\ \ 
        streckung; 
sim[z_] := \ tr[d[z]]; \)

f[z_]  :=  Zhu[sim[InvZhu[z]]];

GeradeAbbilden[-0.5 + 0.2  I, 0.5 + 0.2 I]

KreisAbbilden[tr[0], streckung]

Programm
_______________________________________________________________________

\(\(KreisAbbilden[m_, r_]\  := \ 
      Module\ 
[{}, 
						

        \(g0\  = \ 
          ParametricPlot[

            										{Re[Zhu[m + r \((Cos[w] + I*Sin[w])\)]], \ \ 
              
										Im[Zhu[m + r \((Cos[w] + I*Sin[w])\)]]}, 
            
										{w, 0, 2  π}, 
							
            AspectRatio -> 1, 
							PlotPoints\  -> \ 200, 

            							GridLines\  -> \ Automatic, 
							
            PlotRange\  -> \ {{\(-4\), 4}, {\(-4\), 4}}, 
							
            PlotStyle -> {RGBColor[0, 0, 0]}, 
					
            Prolog\  -> \ Thickness[0.007]

            			\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ]; \)
		
		]; \)\)

\(\(GeradeAbbilden[p_, q_]\  := \ 
      Module\ 
[{}, 
	
			
        \(ParametricPlot[{Re[f[p + t \((q - p)\)]], 
            Im[f[p + t \((q - p)\)]]}, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 

          			\ \ 																				{t, \(-15\), 15}, 
          
										PlotPoints\  -> \ 100, 
										
          AspectRatio -> 1, 
										
          GridLines\  -> \ Automatic, 
										
          PlotRange\  -> \ {{\(-4\), 4}, {\(-4\), 4}}, 
										
          PlotStyle -> {RGBColor[1, 0, 0]}

          		\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ]; \)
\ \ 	
		]; \)\)

_______________________________________________________________________

\(Bildli\  := 
      Module\ [\ \ {g1}, 
\ \ \ \ \ 								
        KreisAbbilden[tr[0.0], streckung]; 
										
        g1\  = \ ParametricPlot[{

              																								\ \ {
                Re[f[0.04  I + t]], Im[f[0.04  I + t]]}, 

              																								\ \ {
                Re[f[0.2  I + t]], Im[f[0.2  I + t]]}, 

              																								\ \ {
                Re[f[0.4  I + t]], Im[f[0.4  I + t]]}, 

              																								\ \ {
                Re[f[0.6  I + t]], Im[f[0.6  I + t]]}, 

              																								\ \ {
                Re[f[0.8  I + t]], Im[f[0.8  I + t]]}, 

              																								\ \ {
                Re[f[I + t]], Im[f[I + t]]}, 

              																								\ \ {
                Re[f[\(-0.04\) I + t]], Im[f[\(-0.04\) I + t]]}, 

              																								\ \ {
                Re[f[\(-0.2\) I + t]], Im[f[\(-0.2\) I + t]]}, 

              																								\ \ {
                Re[f[\(-0.4\) I + t]], Im[f[\(-0.4\) I + t]]}, 

              																								\ \ {
                Re[f[\(-0.6\) I + t]], Im[f[\(-0.6\) I + t]]}, 

              																								\ \ {
                Re[f[\(-0.8\) I + t]], Im[f[\(-0.8\) I + t]]}, 

              																								\ \ {
                Re[f[\(-I\) + t]], Im[f[\(-I\) + t]]}, 

              																								}, {t, \(-5\), 
              5}, 
														PlotPoints\  -> \ 100, 

            														AspectRatio -> 1, 

            														GridLines\  -> \ Automatic, 

            														
            PlotRange\  -> \ {{\(-4\), 4}, {\(-4\), 4}}, 

            														PlotStyle -> {RGBColor[1, 0, 0]}		

            																								\ \ ]; 


        													\ Show[g1, g0]

        																		]\)

\(\(
Bildli\)\)

Options[ParametricPlot]