The Complex Logarithm Function

The complex logarithm is the inverse relationship to the complex exponential function [LINK], and it has quite a complicated Riemann surface. The real part is a surface of revolution of the graph of the (real) exponential function in the plane, while the imaginary part is the right helicoid, resembling a circular staircase (turned on it's side in the image on the right). The real part is multiply covered, which accounts for its unusual coloring.


The movie shows the rotation that takes you between the two views. The multiple-covering of the "horn" that forms the real-part of the logarithm is pulled apart to form the helicoid of the imaginary part.