@Andre, I don't understand why you are not seeing the source notebook. It was attached to two of the previous messages. But here it is.
Gary
Clear["Global`*"]
Manipulate[
GraphicsRow[
{Graphics[{zLine[lnVs[lnPts[m[Theta]]]],ray[maxr],labelZ},Axes->True,Background->White,PlotRange->{{-2pltRatio ,2pltRatio},{-3pltRatio,3pltRatio}}],
If[m[Theta]>Pi, Graphics[{logCrv2[(#-Pi I)&/@logPts[m[Theta]-Pi ]],logRayCrv[maxr],labelW1,labelW2,piUp,piDwn},Axes->True,Background->White,PlotRange->{{-2pltRatio -5,2pltRatio},{-3pltRatio,3pltRatio}}],
Graphics[{logCrv1[logPts[m[Theta]]],logRayCrv[maxr],labelW1,labelW2,piUp,piDwn},Axes->True,Background->White,PlotRange->{{-2pltRatio -5,2pltRatio},{-3pltRatio,3pltRatio}}]]},
Background->White,Frame->All,Spacings->0,ImageSize->Medium],
{{m[Theta],0,Style["[Theta]",Bold,16]},0,2Pi},{{maxr,2,Style["r",Bold,16]},0,r}, FrameLabel->Style["The Logarithm Function",16,Bold],
Initialization :> (
logf[z_]:=Log[Abs[z]]+I Arg[z];
v[z_]:={Re[z],Im[z]};
z[[Theta],r]:=r Exp[I [Theta]];
r = 2;
pltRatio = 1.5;
zPts[[Theta],r]:=Table[r Exp[I x],{x,0,[Theta],.05}];
lnPts[[Theta]_]:= zPts[[Theta],r];
lnVs[pts_] := v[#]&/@pts;
zLine[vectors_] :={Red,Line[vectors]};
rayPts[maxr_]:= Table[z[Pi/4,r],{r,maxr,.001,-.05}];
ray[r_] := {Blue, Line[v[#]&/@rayPts[r]]};
logPts[[Theta]_]:=logf[#]&/@zPts[[Theta],r];
logCrv1[pts_] := {Red,Line[v[#]&/@pts]};
logCrv2[pts_]:={Red,Line[v[#]&/@pts]};
logRayPts[maxr_] := logf[#]&/@rayPts[maxr];
logRayCrv[maxr_] := {Blue,Line[v[#]&/@logRayPts[maxr]]};
labelZ = Inset[Style["z",Bold,12],{-2,-3}];
labelW1 = Inset[Style["w=Log(z)",Bold,12],{-5,-3}];
labelW2 = Inset[Style["Log(z) = ln|z| + i Arg(z)",Bold,8],{-5,-4}];
piUp = Inset[Style["[Pi]",Bold,12],{.5,Pi}];
piDwn =Inset[Style["-[Pi]",Bold,12],{.5,-Pi}]; )
]