Complex Function Plotting Tool

Domain coloring of w = f(z)

Transform

↳ View locked to Riemann. An implicit relation has no single-valued f(z), so the transforms and f/g slots don't apply here.

Parameters

Drag the pad (or type) to set each parameter. The circle is |a| = 1.

Animate  t
t = 0
Parameter sweep

Drag to rotate · scroll to dolly. The north pole is ∞, the south pole is 0.

Left-drag to pan · right-drag to orbit · scroll to zoom. Top-down matches the 2D portrait.

Generator loops (π₁)

Drag to orbit · scroll to zoom. The sheets glue across the branch cut — no false cliff. Values are ≈; the glued topology is exact.

Coloring

For a colorblind-safe figure, pick the Colorblind-safe colormap and check it with the preview above. (The DLMF four-colour map is deliberately not CVD-safe.)

Enhancement
Legend
arg f
|f| = 0large
Cursor readout
z
—
f(z)
—
|f|
—
arg f
—
sheet
—

On the Riemann surface, f(z) is the value of the sheet under the cursor; sheet is its local order (of the sheets over this z) — ≈.

Instruments

Export & share
Suite interop

Drag to pan · scroll to zoom · move the cursor to read f(z). Hue = arg f, brightness = |f|.

What am I looking at?

Each pixel is coloured by w = f(z): the hue is the phase (arg f), and brightness grows with the modulus |f|.

  • Zero — a dark point where all hues meet; the colour wheel winds once per order.
  • Pole — a bright point where the hue winds the opposite way (|f| → ∞).
  • The conformal grid's near-square tiles mark where f preserves angles; they shear or pinch where it doesn't (e.g. at a zero or pole).
  • Grey hatching (with “Flag unreliable regions” on) marks where the phase changes faster than a pixel — near poles and essential singularities — so it reads as an estimate, not resolved structure.
  • A “≈” precision badge appears when f uses a float32-limited special function (ζ strongly, Γ mildly), so its fine detail reads as indicative, not certified.
Monodromy report