Contour Integral

∮C f(z) dz — visualizing phase-weighted accumulation

Function f(z)

Contour

Animation

speed

Running Sum ∮ f(z) dz

Re part 0.000
Im part 0.000
|integral| 0.000
arg(integral) —

Current step

z position —
f(z) —
dz (step) —
contribution f·dz —

Legend

Contour C
Current z on C
f(z) — full vector
Re(f(z)) component
Im(f(z)) component
dz — path direction
f·dz — contribution

What you're seeing

White arrow: f(z) as a full complex vector from z. Blue/pink dashed: its Re and Im components (right-angle decomposition). Teal arrow: dz, the direction of travel along C. Gold arrow (offset): f·dz, the actual contribution added to the integral at this step — it's the complex product of f(z) and dz, so its direction is the sum of their phases.

Watch the gold f·dz arrow for z−1: it always points the same way (upward, Im axis), so contributions never cancel.