Riemann surfaces
## This is a Riemann surface.

##### This image was generated using MATLAB (TM).

This is a projection of the Riemann surface for the cube root function. In
this image, the sheets do not actually intersect except at the origin, which is
known as a *branch point*.

Riemann surfaces arise in the study of complex analytic functions: Because
every complex number has two square roots, three cube roots, and so on, the
cube root function cannot be continuous over all of the complex plane. We call
these functions "multiple-valued," and in order to treat them as continuous
single-valued functions again, it is necessary to replace their
*domains* by the corresponding *Riemann surface*. See
*Complex Analysis* by Lars Ahlfors for a short intuitive treatment of
this topic (at the "cut-and-paste" level). There are more detailed references,
but unfortunately I don't know them offhand.

Last updated on July 22, 1998 by
Kevin Lin.