Lineární funkce

Posunutí pomocí   f: w = z + b

f1[z_,b_]=z+b;
Manipulate[
gr1=ParametricPlot[ReIm[x+I y],{x,-1,1},{y,-1,1},PlotPoints->10,PlotRange->{-2,2},Mesh->10,MeshStyle->{Red,Black}];
gr2=ParametricPlot[ReIm[f1[x+I y,Reb+I Imb]],{x,-1,1},{y,-1,1},PlotPoints->10,PlotRange->{-2,2},Mesh->10,MeshStyle->{Red,Black}];
Show[GraphicsGrid[{{gr1,gr2}}]]
,{Reb,-2,2,0.4},{Imb,-2,2,0.4}]

Stejnolehlost pomocí   f: w = a z,   kde a > 0

f2[z_,a_]=a z;
Manipulate[
gr1=ParametricPlot[ReIm[x+I y],{x,-1,1},{y,-1,1},PlotPoints->10,PlotRange->{-2,2},Mesh->10,MeshStyle->{Red,Black}];
gr2=ParametricPlot[ReIm[f2[x+I y,Rea]],{x,-1,1},{y,-1,1},PlotPoints->10,PlotRange->{-2,2},Mesh->10,MeshStyle->{Red,Black}];
Show[GraphicsGrid[{{gr1,gr2}}]]
,{Rea,0.5,2,0.1}]

Otočení pomocí   f: w = a z,   kde |a|=1

f3[z_,a_]=a z;
Manipulate[
gr1=ParametricPlot[ReIm[x+I y],{x,-1,1},{y,-1,1},PlotPoints->10,PlotRange->{-2,2},Mesh->10,MeshStyle->{Red,Black}];
gr2=ParametricPlot[ReIm[f3[x+I y,Exp[I alpha]]],{x,-1,1},{y,-1,1},PlotPoints->10,PlotRange->{-2,2},Mesh->10,MeshStyle->{Red,Black}];
Show[GraphicsGrid[{{gr1,gr2}}]]
,{alpha,-0.2,3,0.1}]

Kruhová inverze

f: w = 1/(OverBar[z])

f[z_]=1/Conjugate[z]
gr1=RegionPlot[Abs[x+I y]<1,{x,-4,4},{y,-4,4}];
gr2=RegionPlot[Abs[f[u+I v]]<1,{u,-4,4},{v,-4,4}];
Show[GraphicsGrid[{{gr1,gr2}}]]

f[z_]=1/Conjugate[z]
gr0=ContourPlot[Abs[x+I y]==1,{x,-4,4},{y,-4,4},ContourStyle->GrayLevel[0.7]];
gr1=ContourPlot[Abs[x+I y-1]==Abs[x+I y-I],{x,-4,4},{y,-4,4}];
gr2=ContourPlot[Abs[f[u+I v]-1]==Abs[f[u+I v]-I],{u,-4,4},{v,-4,4}];
Show[GraphicsGrid[{{Show[gr0,gr1],Show[gr0,gr2]}}]]

f[z_]=1/Conjugate[z]
gr0=ContourPlot[Abs[x+I y]==1,{x,-4,4},{y,-4,4},ContourStyle->GrayLevel[0.7]];
gr1=ContourPlot[Abs[x+I y+2]==Abs[x+I y],{x,-4,4},{y,-4,4}];
gr2=ContourPlot[Abs[f[u+I v]+2]==Abs[f[u+I v]],{u,-4,4},{v,-4,4}];
Show[GraphicsGrid[{{Show[gr0,gr1],Show[gr0,gr2]}}]]

f[z_]=1/Conjugate[z]
gr0=ContourPlot[Abs[x+I y]==1,{x,-4,4},{y,-4,4},ContourStyle->GrayLevel[0.7]];
gr1=ContourPlot[Abs[x+I y]==Abs[x+I y+1+I],{x,-4,4},{y,-4,4}];
gr2=ContourPlot[Abs[f[u+I v]]==Abs[f[u+I v]+1+I],{u,-4,4},{v,-4,4}];
Show[GraphicsGrid[{{Show[gr0,gr1],Show[gr0,gr2]}}]]

Lineární lomená funkce

f:w=(a z+b)/(c z+d)

f[z_]=z/(z-2)
vv=Solve[f[z]==w,z];
g[w_]=vv[[1,1,2]]
FullSimplify[{f[g[z]],g[f[z]]}]
gr1=RegionPlot[Re[x+I y]<1,{x,-4,4},{y,-4,4}];
gr2=RegionPlot[Re[g[u+I v]]<1,{u,-4,4},{v,-4,4}];
Show[GraphicsGrid[{{gr1,gr2}}]]

f[z_]=(z+1)/(z-1)
vv=Solve[f[z]==w,z];
g[w_]=vv[[1,1,2]]
Manipulate[
gr1=ParametricPlot[ReIm[x+I y],{x,-1+a,a},{y,b,1+b},PlotPoints->10,PlotRange->{-2,2},Mesh->10,MeshStyle->{Red,Black}];
gr2=ParametricPlot[ReIm[g[u+I v]],{u,-1+a,a},{v,b,1+b},PlotPoints->10,PlotRange->{-2,2},Mesh->10,MeshStyle->{Red,Black}];
gr3=ParametricPlot[ReIm[Exp[I t]],{t,-π,π},PlotPoints->10,PlotStyle->Red];
Show[GraphicsGrid[{{gr1,Show[gr2,gr3]}}]]
,{a,0,2,0.1},{b,0,1,0.1}]