A rational function
Two poles at ±i, clipped at the ceiling, and the hue running once round each.
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<notatio-complex-plot-3d value="1/(z^2 + 1)" />Complex Plot 3D
Wolfram's ComplexPlot3D, rendered by <notatio-complex-plot-3d>: |f(z)| as a surface over the complex plane, each face coloured by arg f(z) on the same hue wheel the complex plot paints. A pole is a spike that rises to max-height (default 4) with every hue winding round it; a zero is a dimple the hues wind round the other way. value is notatio; LaTeX goes in a $…$ island.
The surface is sampled on the CPU (samples per side, default 40) through the base package's complex evaluator — the elementary operations and the analytic special functions — or through the engine's own numeric evaluation for anything else; gpu moves the sampling to a compute shader (see GPU evaluation). The view is the one every 3-D figure has: drag to rotate (azimuth / elevation), ctrl/⌘ + wheel to zoom, double-click to reset. Hover to read out (re, im, |f|).
Poles and zeros
Zeros too
The cube roots of unity are the three dimples; the poles are the two spikes. Reading the colour wheel round a point gives its order, and the direction of the winding says which.
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<notatio-complex-plot-3d value="(z^3 - 1)/(z^2 + 1)" domain="-2,2,-2,2" samples="60" />Special functions
The gamma function
Poles at 0, −1, −2, …, each a spike; to the right the surface climbs as the factorials do.
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<notatio-complex-plot-3d value="Gamma(z)" domain="-4,4,-3,3" samples="60" max-height="6" />The Riemann zeta function
The pole at z = 1, and up the critical line the first non-trivial zeros at ½ + 14.13 i, ½ + 21.02 i, ½ + 25.01 i -- the dimples every hue winds round.
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<notatio-complex-plot-3d value="Zeta(z)" domain="-2,3,0,30" samples="70" max-height="3" />GPU evaluation
Set gpu and the grid is evaluated in a WebGPU compute shader through the same complex lowering the portrait uses -- gpu="160" also sets the sample count, since the GPU takes a far denser grid in stride. Where WebGPU is missing, or the expression has no lowering, the CPU sampler runs instead; a GPU badge under the figure says which happened.
A dense rational surface
The rational function again at 160 samples a side, so the spikes come to a point and the hue is continuous across each face.
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<notatio-complex-plot-3d value="(z^3 - 1)/(z^2 + 1)" gpu="160" max-height="3" />An essential singularity
Exp(1/z) near the origin: the surface climbs to the ceiling on the right and falls to the floor on the left, and between them every hue passes infinitely often.
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<notatio-complex-plot-3d value="Exp(1/z)" domain="-1,1,-1,1" gpu="200" max-height="3" elevation="30" />The trigamma function
PolyGamma(1, z) through the analytic kernels on the GPU: double poles at 0, −1, −2, …, so the hue winds twice round each spike.
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<notatio-complex-plot-3d value="PolyGamma(1, z)" domain="-4,2,-2,2" gpu="160" max-height="5" />As an expression
The head is the component: an evaluated ComplexPlot3D(f, (z, a + b i, c + d i)) draws itself, the iterator's complex corners becoming the domain.
Through notatio-out
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<notatio-out format="notatio" value="ComplexPlot3D(1/(z^2 + 1), (z, -2 - 2 * i, 2 + 2 * i))" />Notes
Canvas past 80 samples a side. Below that the surface is SVG polygons, which print and theme through CSS. Above it the same scene is painted on a canvas: a face as a DOM node costs more to parse and lay out than to fill, and at 40 000 faces the difference is a turn of the view that takes a second against one that takes a frame or two.
Clipping, not infinity. |f| near a pole grows without bound; the surface is cut at
max-height, as Wolfram's is, so a single pole does not flatten everything else to the floor.Faces across the branch cut. A face's hue is the circular mean of its corners, so a cell straddling arg = ±π takes the hue between its sides rather than the opposite one.