Hue = arg f(z) (angle, 0→360°). Brightness contours mark |f(z)| = 2ⁿ. Zeros where all colours converge. Poles where colours swirl in reverse.
Scroll / pinch to zoom. Drag to pan. Hover to read z, |f(z)|, arg f(z) at cursor. Save as PNG.
Order-n zero: n rainbow cycles converging. Order-n pole: n cycles diverging. Essential singularity: infinitely many cycles.
Holomorphic f with f'(z)≠0 preserves angles. Locally, image regions look like scaled rotations of the domain.
A complex function f: ℂ → ℂ has a 4-dimensional graph — impossible to visualise directly. Domain colouring (popularised by Frank Farris and Hans Wegert) resolves this by mapping each input point z to a colour determined by f(z): the hue encodes the argument arg(f(z)) ∈ (−π, π] cycling through the full colour wheel, and the brightness encodes |f(z)| using logarithmically spaced contour rings (each ring boundary marks a power of 2 in magnitude).
| Feature | Visual Signature | Example Function |
|---|---|---|
| Simple zero (order 1) | One clockwise rainbow cycle converging to a single point | f(z) = z, f(z) = z−a |
| Zero of order n | n clockwise rainbow cycles converging | f(z) = zⁿ |
| Simple pole | One anti-clockwise rainbow cycle diverging | f(z) = 1/z |
| Pole of order n | n anti-clockwise rainbow cycles | f(z) = 1/zⁿ |
| Essential singularity | Infinite colour oscillation near the point | f(z) = exp(1/z) at z=0 |
| Branch cut | Sharp discontinuity in hue along a ray | f(z) = √z (cut along negative real axis) |
| Conformal region | Locally, grid-squares look like scaled rotations | Any f with f'(z) ≠ 0 |
| Critical point f'(z)=0 | n colours converge without a magnitude zero | f(z) = z² at z=0 (winding 2, no pole) |
| Function | Key Features | Topology |
|---|---|---|
| z² | 1 zero of order 2 at origin; 2 rainbow cycles converge there | 2-to-1 map, double cover |
| z³ | 1 zero of order 3; 3 rainbow cycles; critical point at 0 | 3-to-1 away from 0 |
| sin(z) | Zeros at nπ (n∈ℤ); vertical period 2πi; exponential growth im axis | ∞-to-1; essential sing. at ∞ |
| exp(z) | No zeros; periodic vertically with period 2π; no finite singularities | ∞-to-1; whole function |
| (z−1)/(z+1) | Zero at z=1; pole at z=−1; Möbius transformation | Biholomorphism of ℂ̂ |
| 1/z | Simple pole at 0; conformal everywhere else; inverts unit circle | Automorphism of ℂ̂ |
| z²−1 | Two simple zeros at ±1; no poles; entire function | 2-to-1 outside zeros |
| z·sin(z) | Higher-order zero at 0 (order 2); additional zeros at ±nπ | Entire; exponential type |