Details

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.