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To plot a complex function in a browser, enter an expression such as z^2 in a domain-coloring visualizer. The plot assigns each point in the input complex plane a color based on the function’s output. Typically, hue shows the output’s phase and brightness shows its magnitude—but palettes and controls differ by tool, so check the plot’s legend.
What domain coloring shows
A complex number has a real part and an imaginary part, so both an input z and its output f(z) lie in a plane. A conventional graph of a complex function would need four real dimensions: two for the input and two for the output. A flat image cannot show that graph directly. Domain coloring instead colors each sampled input point according to its function value.
At each screen pixel, a visualizer identifies a point in the displayed complex plane, evaluates the function there, and converts the result into a color. The Interactive Mathematics explanation of domain coloring and the Complex Function Visualizer illustrate common versions of this approach.
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A common encoding maps the argument, or phase, of f(z) to hue and its modulus, or magnitude, to brightness or value. In one illustrated convention, positive real values are red and negative real values are cyan. That is an example, not a universal color standard. Magnitude scaling also varies; use the visualizer’s legend and settings to interpret the image rather than assuming another plot uses the same palette.
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A pure phase portrait emphasizes direction and may hide differences in magnitude. Enhanced portraits add contour lines for phase and modulus, making changes in the function easier to compare. The Interactive Mathematics treatment describes these contours and provides an interactive applet.
Plot a function in a browser
- Open an interactive plotter. The Complex Function Visualizer accepts expressions in z and also supports time-dependent functions using t. The jcponce Complex Function Plotter is another browser-based option.
- Enter a supported expression. Try
z,1/z,z^2,e^z, orsin(z). Syntax and available functions are specific to each tool: consult its examples or documentation if an expression is rejected. - Inspect the display and its controls. Identify which colors represent phase and how magnitude is mapped. If available, enable phase and modulus contours, adjust the plotted region, or use animation controls for expressions involving t.
- Check special points and conventions. Look for places where the function is zero or undefined, and check whether a branch cut or excluded domain affects the image. A sharp seam can be mathematically meaningful, not a rendering defect.
How browser plotters differ
| Tool | Documented approach and features | What to check |
|---|---|---|
| jcponce Complex Function Plotter | Its README describes a WebGL plotter: expressions are parsed with PEG.js and transformed into WebGL-compatible code for a shader template. It documents examples, modulus and phase level curves, and functions including trigonometric and hyperbolic functions, conjugate, modulus, powers, logarithm, exponential, real and imaginary parts, gamma, zeta, and Joukowsky. | Use its README for the expression syntax and supported functions. These capabilities describe this plotter, not browser plotters generally. |
| Complex Function Visualizer | An interactive browser page with example expressions including z^2, 1/z, e^z, ln(z), cos(z), and zeta(z). It supports time-dependent functions using t. |
Check the page’s controls and display to see how it handles the expression, palette, magnitude, and animation. |
| LK Forge’s 2026 implementation | The article describes evaluating complex arithmetic and painting sampled output onto a 2D canvas without WebGL. It reports about 3.4 million evaluations per second and 200,000 evaluations in 58 ms on a single thread for its rational-function example. | Those are publisher-reported, implementation-specific figures, not a performance guarantee for other devices or plotters. |
Rendering method alone does not tell you which plotter will feel faster on your device. Expression support, contour and color controls, animation, and the plotted region all affect whether a tool suits a particular exploration. Check the selected tool’s current documentation and try a simple expression before relying on a more complex one.
Read zeros, poles, and branch cuts
Zeros and poles
A zero at z0 means f(z0) = 0. A pole is a point where the function is undefined and its magnitude grows without bound as the point is approached. In common phase-and-magnitude portraits, a zero often appears as a dark center around which hue cycles; a pole often appears as a bright center with a phase cycle. These are visual clues, not a substitute for checking the formula.
- For
f(z)=z, the origin is a zero. - For
g(z)=1/z, the origin is a pole and the function is undefined there. - For
(z-1)/(z^2+z+1), there is a zero at 1 and poles at (−1 ± √3i)/2, as described in the Interactive Mathematics examples.
Modulus contours help distinguish the behavior around zeros and poles; phase contours reveal how the output’s direction changes. A finite-resolution image can miss an exceptional point or make nearby features hard to separate, so verify candidate locations algebraically.
Branch cuts and excluded points
Some functions require a branch convention. For example, a principal-logarithm plot can show a seam along the negative real axis. That seam reflects the chosen branch of the logarithm, not necessarily a rendering fault; the LK Forge explanation discusses this example. When interpreting a plot, note the branch convention and any points or regions where the function is undefined.
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A striking pattern is evidence to investigate, not proof of a property such as analyticity. The Interactive Mathematics discussion warns that without constraints such as continuity or differentiability, isochromatic sets can be arbitrary. For analytic functions, a pure phase portrait almost uniquely determines the function, but that relationship does not generally hold for arbitrary functions; distinct analytic and non-analytic examples can share a phase portrait away from zeros and poles.
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- Use phase and modulus contours when magnitude or local variation matters.
- Check the function’s domain, zeros, poles, and branch convention independently.
- State the plotted region and the palette and magnitude mapping when sharing or comparing an image.
- Treat apparent features cautiously: pixel sampling and display limits do not establish that every zero or pole has been found.
For a deeper treatment of phase portraits, DomainColoring.jl documentation cites Elias Wegert’s Visual Complex Functions: An Introduction with Phase Portraits (2012) as an inspiration for its plots. It is supplementary reading, not a requirement for using a browser plotter.
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