       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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