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).