(*Mathematica code to draw images of unit disk*)ctor[z_]:=(output={};
Do[AppendTo[output,{Re[z[[k]]],Im[z[[k]]]}],{k,1,Length[z]}];
output)

plotpolylistCA[vert_]:=(vert3={};
Do[vert1=ctor[vert[[k]]];
AppendTo[vert3,vert1];,{k,1,Length[vert]}];
plotpolylist[vert3])

plotpolylist[vertlist_]:=Show[Table[Graphics[{(*GrayLevel[1],Thickness[.01],{Polygon[vertlist[[k]]]},*)GrayLevel[If[k==Length[vertlist],0,.1]],Thickness[If[k==Length[vertlist],.005,.005]],{Line[vertlist[[k]]]}},{PlotRange->All,Axes->None,AspectRatio->Automatic}],{k,1,Length[vertlist]}]]

makeradial:=(num0=16;
angles=Table[2 Pi k/num0,{k,1,num0}];
out={};
lambda1=.9;
inner=.1;
num=Ceiling[Log[inner]/Log[lambda1]];
radpts=Table[lambda1^k,{k,1,num}];
AppendTo[radpts,0];
PrependTo[radpts,1];
Do[(AppendTo[out,Exp[I angles[[k]]] radpts];),{k,1,Length[angles]}];
lambda2=.8;
inner2=.25;
num=Ceiling[Log[inner2]/Log[lambda2]];
radpts3={.25,.5,.75,.9,.95,.99};
Do[(num=5 Ceiling[2 Pi/(1-radpts3[[k]])];
AppendTo[out,radpts3[[k]] Table[Exp[2 Pi I k/num],{k,0,num}]];),{k,1,Length[radpts3]}];
AppendTo[out,Table[Exp[2 Pi I k/100],{k,0,100}]])

makeradial;

plotpolylistCA[out]

f[z_]:=z+(z^2)/2;
plotpolylistCA[f[out]]

f[z_]:=z+(z^8)/8;
plotpolylistCA[f[out]]

plotpolylistCA[Exp[out]]

Manipulate[plotpolylistCA[out + a out^2 + b  out^3],{a,-1,1},{b,-2,2}]

koebefcn[z_]:=z + 2 z^2 + 3z^3 + 4 z^4 +5 z^5 +6 z^6;
plotpolylistCA[koebefcn[out]]