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ComplexPlot

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

ComplexPlot[f, {z, zmin, zmax}, opts...]

Domain-colouring plot of the complex function f over the rectangular region in the complex plane with corners zmin and zmax. z is bound to Complex[x, y] at each grid point; f must return a complex or real number. Each cell's hue encodes Arg(f(z)) on the cyclic "Cyclic" ramp, and |f(z)| sets an HSL lightness: zeros fade to black, poles to white. ComplexPlot is HoldAll: f and the iterator spec are held unevaluated until z is given a numeric complex value. Options: PlotPoints grid resolution per axis (default 400) ColorFunction a named ramp string, or a function of the eight Mathematica arguments Re[z], Im[z], Abs[z], Arg[z], Re[f], Im[f], Abs[f], Arg[f] (so #8 is the phase of the value). Named ramps: "PhaseRings" (hue=phase, brightness=log|w| rings — highlights poles and zeros), "Cyclic", "Rainbow", "CoolTones", "WarmTones", "Greyscale", "Temperature" ColorFunctionScaling True (default): scale each of the eight arguments to [0,1] across the sampled domain before calling a custom ColorFunction RegionFunction f[x,y] mask; excluded cells are not drawn PlotLegends Automatic / True: attach a vertical phase color scale bar (thermal ramp, -π at bottom, π at top) Standard Graphics options (Axes, AspectRatio→1, Frame, PlotRange, AxesLabel, GridLines, ImageSize, Background, PlotLabel, …) pass through to the Graphics[...] result. Examples: ComplexPlot[z^2, {z, -2-2I, 2+2I}] ComplexPlot[Sin[z], {z, -Pi-Pi*I, Pi+Pi*I}] ComplexPlot[1/(z^2+1), {z, -2-2I, 2+2I}, PlotPoints->80] ComplexPlot[(z^2+1)/(z^2-1), {z, -2-2I, 2+2I}, PlotLegends->Automatic]

Examples

No verified examples yet for this function.

Algorithm

complexplot.c — ComplexPlot and ComplexPlot3D.

Both functions share the same domain-parsing logic and option set. The only structural difference is what they build from the evaluated grid: ComplexPlot emits Rectangle primitives into Graphics[], and ComplexPlot3D emits Polygon quads (height = |w|, colour = arg(w)) into Graphics3D[].

Coloring convention: the default maps the phase arg(w) onto the cyclic "Cyclic" ramp (t = (atan2(im, re) + π) / (2π) ∈ [0, 1]) and folds the modulus in as an HSL lightness — zeros fade to black, poles to white — so ComplexPlot[f] and ComplexPlot[f, ColorFunction -> "Cyclic"] are identical. A custom ColorFunction receives the eight Mathematica arguments

  Re[z], Im[z], Abs[z], Arg[z], Re[f], Im[f], Abs[f], Arg[f]

(so #8 is the phase of the value); with ColorFunctionScaling→True (default) each is scaled to [0,1] across the sampled domain.

Both are HoldAll: the body and the iterator spec are held unevaluated until z is bound to Complex[x, y] at each grid point.

Domain spec:

  {z, zmin, zmax}  — z is the complex iterator variable; zmin and zmax
  are evaluated and their Re/Im parts define the rectangular plotting
  domain: xmin=Re(zmin), xmax=Re(zmax), ymin=Im(zmin), ymax=Im(zmax).
  Both endpoints may be real (imaginary part = 0).

Implementation notes

Attributes: HoldAll, Protected.

References

See also: HoldAll, Plot, Arg, AspectRatio, Frame, ImageSize