Complex Maps

<<Graphics`ComplexMap`

SetOptions[Graphics,BaseStyle->{FontSize->20,FontWeight->"Bold"}];

Remove["Global`*"]

Conformal[mapType_,func_,xrange_,yrange_,options___]:=Framed[GraphicsArray[{mapType[#&,xrange,yrange,options],mapType[func,xrange,yrange,options]},ImageSize->500,PlotLabel->Style[func[z],Bold,16,FontFamily->"Helvetica"]]];
myf[z_]:=1/(z+1);

Conformal[CartesianMap,myf,{0.5,10,0.5},{-5,5,0.25},AxesLabel->{x,y},AxesOrigin->{0,0}]

myf[z_]:=(z-I)/(z+I);

Conformal[CartesianMap,myf,{-5,5},{0,5},Lines->30]

myf[z_]:=I*(1+z)/(1-z);

Options[Conformal]

Conformal[PolarMap,myf,{0,0.99},{0,2 Pi},Lines->20,PlotRange->{{-2,2},{-2,2}}]

myf[z_]:=(I*(1+z)/(1-z));
Conformal[PolarMap,myf,{0,0.99},{0,2 Pi},Lines->20,PlotRange->{{-2,2},{-2,2}}]
?Conformal

myTa[a_,z_]:=(z-a)/(1-Conjugate[a]*z)
a=0.65
myT[z_]= myTa[a,z]
Conformal[PolarMap,myT,{0,0.99},{0, Pi},Lines->20,PlotRange->{{-1.2,1.2},{-1.2,1.2}}]

myf2[z_]:=Exp[z/Abs[z]^2]
Conformal[ PolarMap, myf2,{2,4.99},{0, Pi},Lines->20,PlotRange->{{-5,5},{0,5}}]

myf3[z_]:=Exp[z]
Conformal[ PolarMap, myf3,{2,4.99},{0, Pi},Lines->20,PlotRange->{{-5,5},{0,5}}]

Conformal[ CartesianMap, myf3,{2,4.99},{0, Pi},Lines->20,PlotRange->{{2,5},{0,5}}]

Plot3D[Im[ArcSin[(x+I y)^4]],{x,-2,2},{y,-2,2},Mesh->None,PlotStyle->Directive[Yellow,Specularity[White,20],Opacity[0.8]],ExclusionsStyle->{None,Red}]

$InverseTrigFunctions={ArcSin,ArcCos,ArcSec,ArcCsc,ArcTan,ArcCot,ArcSinh,ArcCosh,ArcSech,ArcCsch,ArcTanh,ArcCoth};
Table[Plot3D[Im[f[(x+I y)^3]],{x,-2,2},{y,-2,2},PlotStyle->Directive[Pink,Specularity[White,50],Opacity[0.8]],ExclusionsStyle->{None,Red},ClippingStyle->None,Mesh->None,PlotLabel->Im[f[(x+I y)^3]],Ticks->None],{f,$InverseTrigFunctions}]

(* see also http://math-www.uni-paderborn.de/~axel/graphs/ *)

DensityPlot[Arg[Sqrt[x+I y]],{x,-3,3},{y,-3,3},Mesh->10,MeshFunctions->{Re[Sqrt[#1+I #2]]&,Im[Sqrt[#1+I #2]]&}]

ContourPlot[Abs[JacobiSN[ux+I uy,1/3]],{ux,0,4 EllipticK[1/3]},{uy,0,4 EllipticK[2/3]}]