Building on Heike's ColorFunction, I came up with this:
z = Transpose@Reverse@Sin@
Outer[Complex, Range[-Pi, Pi, 0.01], Range[-Pi, Pi, 0.01]];
hsbdata = Transpose[{
Rescale[Arg[z], {-Pi, Pi}],
1 - 0.05/Abs[Sin[2 Pi Abs[z]]],
0.02/Abs[Sin[2 Pi Abs[z]]] + Abs[Sin[2 Pi Im@z] Sin[2 Pi Re@z]]^0.25}
, {3, 1, 2}];
Image[hsbdata, ColorSpace -> "HSB"]
The white bits are the trickiest - you need to make sure the brightness is high where the saturation is low, otherwise the black lines appear on top of the white ones.
EDIT
I wanted to tidy up the code and make it easy to produce domain colouring plots of other functions.
The functions defined below are:
complexGrid[max,n] simply generates an $n\times n$ grid of complex numbers ranging from $-max$ to $+max$ in both axes.
complexHSB[Z] takes an array $Z$ of complex numbers and returns an array of $\{h,s,b\}$ values. I've tweaked the colour functions slightly. The initial $\{h,s,b\}$ values are calculated using Heike's formulas, except I don't square $s$. The brightness is then adjusted so that it is high when the saturation is low. The formula is almost the same as $b2=\max (1-s,b)$ but written in a way that makes it Listable.
domainImage[func,max,n] calls the previous two functions to create an image. func is the function to be plotted. The image is generated at twice the desired size and then resized back down to provide a degree of antialiasing.
domainPlot[func,max,n] is the end user function. To put tick marks and labels on the final plot I just create an empty ContourPlot and embed the domain colouring image within it. Ideally this should take options which can be passed to ContourPlot but I've never got to grips with option handling so I just put in labels and styles which work for me.
complexGrid =
Compile[{{max, _Real}, {n, _Integer}},
Module[{r}, r = Range[-max, max, 2 max/(n - 1)];
Outer[Plus, -I r, r]]];
complexHSB =
Compile[{{Z, _Complex, 2}},
Module[{h, s, b, b2}, h = 0.5 + Arg[Z]/(2 Pi);
s = Abs[Sin[2 Pi Abs[Z]]];
b = Abs[Sin[2 Pi Im[Z]] Sin[2 Pi Re[Z]]]^0.25;
b2 = 0.5 ((1 - s) + b + Sqrt[(1 - s - b)^2 + 0.01]);
Transpose[{h, Sqrt[s], b2}, {3, 1, 2}]]];
domainImage[func_, max_, n_] :=
ImageResize[
ColorConvert[
Image[complexHSB@func@complexGrid[max, 2 n], ColorSpace -> "HSB"],
"RGB"], n];
domainPlot[func_, max_: Pi, n_: 500] :=
ContourPlot[0, {x, -max, max}, {y, -max, max}, Contours -> {},
RotateLabel -> False,
FrameLabel -> {"Re[z]", "Im[z]",
"Domain Colouring of " <> ToString@StandardForm@func@"z"},
BaseStyle -> {FontFamily -> "Calibri", 14},
Epilog ->
Inset[domainImage[func, max, n], {0, 0}, {Center, Center},
2` max]];
Examples follow:
It's informative to plot the untransformed complex plane to understand what the colours indicate:
domainPlot[#&];
A simple example:
domainPlot[Sqrt]
Plotting a pure function:
domainPlot[(#+I)/(#-1)&]
I think this one is very pretty:
domainPlot[Log]
