A Riemann surface is a surface that represents the domain of a multiple-valued complex function.

A Riemann surface is a surface-like configuration that covers the complex plane with several, and in general infinitely many, "sheets." These sheets can have very complicated structures and interconnections. Riemann surfaces are one way of representing multiple-valued functions; another is branch cuts. The above plot shows Riemann surfaces for solutions of the equation [w(z)]^d + w(z) + z^(d - 1) = 0 with d = 2, 3, 4, and 5, where w(z) is the Lambert W-function (M. Trott).

branch cut | function field | ideal | ring

graduate school level

Bernhard Riemann