Riemann Surfaces & Branch Points

Multi-valued complex functions like √z and log(z) become single-valued on their Riemann surfaces — stacked "sheets" connected at branch cuts. As you loop around the branch point at z=0, the function transitions between sheets. The color encodes argument (hue) and modulus (brightness) via domain coloring.

Function

Sheet: 0 Branch cut angle (°): 180 Zoom: 2.0

Cursor z: —

f(z): —

|f(z)|: —

arg(f(z)): —

√z: 2 sheets, branch point z=0
Loop once → sheet 0↔1

log(z): ∞ sheets, same branch pt
Each loop adds 2πi to imaginary part

z^(1/3): 3 sheets
Loop once → cyclic permutation

Monodromy group encodes sheet-swapping topology