@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.
P. S. This code has been edited so it can now be cut and pasted into a notebook, where it should produce a plot.
Gary
Clear["Global`*"]
Manipulate[
GraphicsRow[
{Graphics[{zLine[lnVs[lnPts[mTheta]]],ray[maxr],labelZ},Axes->True,Background->White,PlotRange->{{-2pltRatio ,2pltRatio},{-3pltRatio,3pltRatio}}],
If[mTheta>Pi,
Graphics[{logCrv2[(#-Pi I)&/@logPts[mTheta-Pi ]],logRayCrv[maxr],labelW1,labelW2,piUp,piDwn},Axes->True,Background->White,PlotRange->{{-2pltRatio -5,2pltRatio},{-3pltRatio,3pltRatio}}],
Graphics[{logCrv1[logPts[mTheta]],logRayCrv[maxr],labelW1,labelW2,piUp,piDwn},Axes->True,Background->White,PlotRange->{{-2pltRatio -5,2pltRatio},{-3pltRatio,3pltRatio}}]]},
Background->White,Frame->All,Spacings->0,ImageSize->Medium],
{{mTheta,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}];
)
]