Principal Branch

Author

Eric W. Weisstein
April 15, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/ComplexAnalysis/PrincipalBranch.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/PrincipalBranch.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Plots for ln z

<<Utilities`Typesetting`

branches=Show[Graphics3D[ParametricPlot3D[Evaluate[Table[{r Cos[θ],r Sin[θ],Im[Log[r Exp[I θ]]]+2Pi k},{k,-1,1}]],{r,.01,1},{θ,0,2π},
DisplayFunction->Identity,
PlotPoints->{4,25}][[1]]/.Polygon[x_]:>If[Max[z=Last/@x]-Min[z]>.5,Polygon[Join[#-{0,0,2Pi}&/@Take[x,2],Take[x,-2]]],Polygon[x]]],
Axes->Automatic,
AxesLabel->TraditionalForm/@{x,y,HoldForm[Im[Log[z]]]},
AxesEdge->{{-1,-1},{1,-1},{1,1}},
BoxRatios->{1,1,1},
PlotLabel->StyleForm["multiple branches",FontSlant->"Italic",FontFamily->"Times"],
PlotRange->{-10,10}]

principal=Show[Graphics3D[ParametricPlot3D[{r Cos[θ],r Sin[θ],Im[Log[r Exp[I θ]]]},{r,.01,1},{θ,0,2π},
DisplayFunction->Identity,
PlotPoints->{4,25}][[1]]/.Polygon[x_]:>If[Max[z=Last/@x]-Min[z]>.5,Polygon[Join[#-{0,0,2Pi}&/@Take[x,2],Take[x,-2]]],Polygon[x]]],
Axes->Automatic,
AxesLabel->TraditionalForm/@{x,y,HoldForm[Im[Log[z]]]},
AxesEdge->{{-1,-1},{1,-1},{1,1}},BoxRatios->{1,1,1},
PlotLabel->StyleForm["principal branch",FontSlant->"Italic",FontFamily->"Times"],
PlotRange->{-10,10}]

Show[GraphicsArray[{branches,principal}]]