Domain coloring of multi-valued complex functions — hue=argument, brightness=|f(z)|
About
Multi-valued complex functions like √z and log z cannot be made single-valued on ℂ. A branch cut is a line where we "cut" the domain. The Riemann surface is the multi-sheeted manifold where the function IS single-valued.
Hover over canvas to see f(z)
Domain Coloring
Hue = arg(f(z)) ∈ [0, 2π) Brightness = log|f(z)| (periodic lattice) Branch cut: white line where argument jumps discontinuously
Zeros: dark spots (all colors meet)
Poles: bright spots
Key Idea
Going around the branch point (z=0 for √z), arg(z) increases by 2π, so arg(√z) increases by π — you end up on the other sheet. The Riemann surface for √z has 2 sheets; for z^(1/3) it has 3.