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MathGroup Archive 2007

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Re: Deriving parametric plot of a branch cut

  • To: mathgroup at smc.vnet.net
  • Subject: [mg79943] Re: Deriving parametric plot of a branch cut
  • From: chuck009 <dmilioto at comcast.com>
  • Date: Thu, 9 Aug 2007 05:22:05 -0400 (EDT)

I'm following up on this thread.  With the nice Exclusion code provided by version 6, I can easily plot two logarithmic sheets of the imaginary surface for f[z]=Log[1+1/(e^z Sqrt[z]).  Note that each logarithmic sheet bifurcates into two square-root sheets.  I like the nice interactive 3-D features of version 6.  

wSheet1 = Log[1 + 1/(Exp[z]*Sqrt[z])]; 
wSheet1b = Abs[1 + 1/(Exp[z]*Sqrt[z])] + 
       I*(Arg[1 + 1/(Exp[z]*Sqrt[z])] + 2*Pi); 
wSheet2 = Log[1 + 1/(Exp[z]*Abs[z]^(1/2)*
              Exp[(I/2)*(Arg[z] + 2*Pi*I)])]; 
wSheet2b = 
     Abs[1 + 1/(Exp[z]*Abs[z]^(1/2)*Exp[(I/2)*(Arg[z] + 2*Pi*I)])] + 
       I*(Arg[1 + 1/(Exp[z]*Abs[z]^(1/2)*
                     Exp[(I/2)*(Arg[z] + 2*Pi*I)])] + 2*Pi); 
xmax = 15; 
pSheet1 = Plot3D[Evaluate[Im[wSheet1 /. z -> x + I*y]], {x, -4, 4}, 
       {y, -xmax, xmax}, AspectRatio -> 1, Mesh -> All, 
       PlotRange -> {{-xmax, xmax}, {-xmax, xmax}, {-xmax, xmax}}, 
       BoxRatios -> {1, 1, 1}, ClippingStyle -> None, 
   PlotPoints -> 25, 
       PlotStyle -> Red]; 
pSheet1b = Plot3D[Evaluate[Im[wSheet1b /. z -> x + I*y]], 
       {x, -4, 4}, {y, -xmax, xmax}, AspectRatio -> 1, Mesh -> All, 
       PlotRange -> {{-xmax, xmax}, {-xmax, xmax}, {-xmax, xmax}}, 
       BoxRatios -> {1, 1, 1}, ClippingStyle -> None, 
   PlotPoints -> 25, 
       PlotStyle -> Green]; 
pSheet2 = Plot3D[Evaluate[Im[wSheet2 /. z -> x + I*y]], {x, -4, 4}, 
       {y, -xmax, xmax}, AspectRatio -> 1, Mesh -> All, 
       PlotRange -> {{-xmax, xmax}, {-xmax, xmax}, {-xmax, xmax}}, 
       BoxRatios -> {1, 1, 1}, ClippingStyle -> None, 
   PlotPoints -> 25, 
       PlotStyle -> Blue]; 
pSheet2b = Plot3D[Evaluate[Im[wSheet2b /. z -> x + I*y]], 
       {x, -4, 4}, {y, -xmax, xmax}, AspectRatio -> 1, Mesh -> All, 
       PlotRange -> {{-xmax, xmax}, {-xmax, xmax}, {-xmax, xmax}}, 
       BoxRatios -> {1, 1, 1}, ClippingStyle -> None, 
   PlotPoints -> 25, 
       PlotStyle -> Yellow]; 
Show[{pSheet1, pSheet1b, pSheet2, pSheet2b}, Boxed -> False, 
   Axes -> False]






> chuck009 wrote:
> > I've been working on an interesting problem of
> late:  f[z_]:=Log[1+Exp[-z]/Sqrt[z]] posted in here
> earlier.  I'm unable to plot the Riemann surface of
> this function even with the nice code written by
> Michael Trott (his code fails to handle the branch
> cuts).


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