Question

I’m currently reading the Mathematica Guidebooks for Programming and I was trying to work out one of the very first program of the book. Basically, when I run the following program:

Plot3D[{Re[Exp[1/(x + I y)]]}, {x, -0.02, 0.022}, {y, -0.04, 0.042}, 
 PlotRange -> {-1, 8}, PlotPoints -> 120, Mesh -> False, 
 ColorFunction -> Function[{x1, x2, x3}, Hue[Arg[Exp[1/(x1 + I x2)]]]]]

either I get a 1/0 error and e^\infinity error or, if I lower the PlotPoints options to, say, 60, an overflow error. I have a working output though, but it’s not what it’s supposed to be. The hue seems to be diffusing off the left corner whereas it should be diffusing of the origin (as can be seen on the original output)

Here is the original program which apparently runs on Mathematica 5 (Trott, Mathematica Guidebook for Programming):

Off[Plot3D::gval];

Plot3D[{Re[Exp[1/(x + I y)]], Hue[Arg[Exp[1/(x + I y)]]]},
       {x, -0.02, 0.022}, {y, -0.04, 0.042},
       PlotRange -> {-1, 8}, PlotPoints -> 120, Mesh -> False]

Off[Plot3D::gval];

However, ColorFunction used this way (first Plot3D argument) doesn’t work and so I tried to simply adapt to its new way of using it.

Well, thanks I guess!

Answer 1

I finally found two alternative ways to solve my problem. The first one is to simply use the << Version5`Graphics` command to use Plot3Dfunction the way it worked with Mathematica V5. The code taken from the book works just like it used to.

However, if one wishes to display correctly the hue (that is, without diffusion off the left-hand corner) with the latest version, the Rescale function must be used, just like this:

Plot3D[Evaluate[Re[f[x, y]]], {x, -.02, .022}, {y, -0.04, 0.042},
PlotRange -> {-1, 2}, PlotPoints -> 120, Mesh -> False,
ColorFunction -> Function[{x, y, z}, Hue@Rescale[Arg[f[x, y]], {-π, π}]],
ColorFunctionScaling -> False,
ClippingStyle -> None]

I suppose the argument function in Mathematica does not map automatically to the [-Pi,Pi) range and so it must be rescaled to this domain. The result is quite good-looking, although there are some minor differences with the original plot.