If we attempt to define the logarithm of a complex number as some kind of ‘inverse’ to the exponential function, we have to face the fact that the latter is not a bijection, hence does not have an inverse in the technical sense. Unlike the real case, there is no very natural way to restrict its domain and codomain in such a way that it becomes a bijection – although a variety of more or less artificial such choices exist (such as the ‘cut plane’ ℂn below) and are indeed useful.

In classical terms, the logarithm has to be ‘multivalued’. The way in which its multiplicity of values fit together is closely analogous to the way that the measurement of an angle by radians gives not a single real number, but an infinite list differing only by multiples of 2π.

We shall discuss these ideas below, and apply them to a topological invariant known as the winding number of a curve relative to a point. In essence the total angle traversed by a point on the curve is measured as it moves continuously from one end to the other: if divided by 2π this gives the number of times that the curve winds around the point in question. This concept is extremely useful in the deeper parts of the subsequent theory.

Radian measure of angles

We first relate the ‘power series’ definition of sine to the usual geometric one, with the angle being measured in radians.