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ComplexPlot3D

Status: Experimental

present and registered, but lightly documented and not yet covered by dedicated tests.

Description

ComplexPlot3D[f, {z, zmin, zmax}, opts...]

Three-dimensional surface plot of a complex function: height = |f(z)|, colour = Arg(f(z)) via the thermal ramp (same default as Plot3D). z is bound to Complex[x, y] at each grid point; the result is a Graphics3D[...] object rendered in an orbit-camera window. ComplexPlot3D is HoldAll. Options: PlotPoints grid resolution per axis (default 200) ColorFunction a named ramp string, or a function of the eight arguments Re[z], Im[z], Abs[z], Arg[z], Re[f], Im[f], Abs[f], Arg[f] (see ComplexPlot); "PhaseRings" recommended ColorFunctionScaling True (default): scale the eight args to [0,1] RegionFunction f[x,y] mask PlotLegends Automatic / True: attach a vertical phase color scale bar (-π at bottom, π at top) Lighting -> None disables Lambertian shading (recommended for accurate phase colours; default Automatic) Standard Graphics3D options pass through to the result. Examples: ComplexPlot3D[z^2, {z, -2-2I, 2+2I}] ComplexPlot3D[Sin[z], {z, -2-2I, 2+2I}, Lighting->None] ComplexPlot3D[1/z, {z, -2-2I, 2+2I}, PlotLegends->Automatic, Lighting->None]

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: Plot3D, HoldAll, ComplexPlot, Lighting