Branch Point - Transcendental and Logarithmic Branch Points

Transcendental and Logarithmic Branch Points

Suppose that g is a global analytic function defined on a punctured disc around z₀. Then g has a transcendental branch point if z₀ is an essential singularity of g such that analytic continuation of a function element once around some simple closed curve surrounding the point z₀ produces a different function element. An example of a transcendental branch point is the origin for the multi-valued function

for some integer k > 1. Here the monodromy around the origin is finite.

By contrast, the point z₀ is called a logarithmic branch point if it is impossible to return to the original function element by analytic continuation along a curve with nonzero winding number about z₀. This is so called because the typical example of this phenomenon is the branch point of the complex logarithm at the origin. Going once counterclockwise around a simple closed curve encircling the origin, the complex logarithm is incremented by 2πi. Encircling a loop with winding number w, the logarithm is incremented by 2πi w.

There is no corresponding notion of ramification for transcendental and logarithmic branch points since the associated covering Riemann surface cannot be analytically continued to a cover of the branch point itself. Such covers are therefore always unramified.