Log Function on a Riemann Surface

Pantea Ferdosian
Log Function on a Riemann Surface
Math 401; Mathematics Through 3D Printing
George Mason University

Gauss’s student, Bernhard Riemann, made some powerful insights in the mid 19th century. The first part of Riemann’s contribution is the idea that we need more than two complex planes to visualize some functions involving i.


A Riemann Surface can help us think in 4 dimensions. Riemann’s big idea is that the domain of the input values of our multi-function should not be a flat 2 dimensional plane. Our domain should instead be a curved surface living in higher dimensional space, a Riemann surface.

A Riemann surface is a geometric object glued from open subsets of the complex plane using holomorphic maps. They are of fundamental importance throughout geometry, analysis, and algebra. For example, it is a remarkable fact that every Riemann surface can be described as a complex algebraic curve, that is, the locus of zeroes of a polynomial in two complex variabl