Panel 1 Domain coloring of Log z
Each point of the plane is colored by its value of Log z: hue = Arg z (the imaginary part) and brightness = ln|z| (the real part — dark near 0, bright far out). Watch the negative real axis: the hue snaps from one color straight to its opposite. That hard seam is the discontinuity of Log z.
○ at the origin: Log 0 is undefined — a logarithmic singularity. The same point z is also live in Panels 2 & 3.
Panel 2 Interactive point probe
Click or drag anywhere to place z. The readout gives its rectangular and polar forms, its principal Arg z, and three of its infinitely many log values (k = −1, 0, 1), which differ by 2πi. The principal value is highlighted.
Drag z slowly across the negative real axis and watch Arg z flip sign — the readout jumps by 2π.
Panel 3 Walk around the origin — where continuity breaks
A point travels counterclockwise on a circle of radius r. On the right, Arg z is plotted against the angle traveled. It climbs smoothly from −π to +π — then, the instant the point crosses the negative real axis, it jumps straight back to −π. The function can't stay continuous there. That forced jump is exactly why a branch cut is needed.
Panel 4 The Riemann surface — the log's “true” domain
The jump in Panel 3 is an artifact of forcing one value per point. The honest picture is a helicoid: height = Im(log z) = the argument, which keeps increasing by 2π each lap around the origin instead of resetting. Each full turn is a different sheet (a different k). The log is perfectly continuous up here — drag to rotate. The green band is the principal sheet (k = 0); the branch cut is just the edge where we slice one sheet off to flatten it back into Panel 1.
Drag horizontally to spin · the vertical seams between colors are the θ = ±π edges where consecutive sheets join.