Riemann Surface of the Logarithm
The Riemann surface of the logarithm is made up of infinitely many interconnected sheets covering the complex plane (right graphic) corresponding to horizontal strips of height in the complex plane (left graphic).
The function is not single-valued over the whole complex plane, but it is single-valued in any region D that does not contain two points , such that . An example of such a region is the horizontal strip or any translation of D along the imaginary axis. Consider the complex plane as the union of the pairwise disjoint strips . The image of each strip , , is a complex plane with a slit along the non-negative part of the real axis. With , these planes can be arranged so that:
• the plane is situated over ;
• the directions of their real and imaginary axes are all the same;
• their origins are on the line orthogonal to the plane at its origin;
• the front edge of a slit of the plane is identified with the back edge of the slit of the plane.
The surface constructed in this way is the Riemann surface of the logarithm . The function is single-valued over the whole complex plane.