Branch Point - Riemann Surfaces

Riemann Surfaces

The concept of a branch point is defined for a holomorphic function ƒ:X → Y from a compact connected Riemann surface X to a compact Riemann surface Y (usually the Riemann sphere). Unless it is constant, the function ƒ will be a covering map onto its image at all but a finite number of points. The points of X where ƒ fails to be a cover are the ramification points of ƒ, and the image of a ramification point under ƒ is called a branch point.

For any point P ∈ X and Q = ƒ(P) ∈ Y, there are holomorphic local coordinates z for X near P and w for Y near Q in terms of which the function ƒ(z) is given by

for some integer k. This integer is called the ramification index of P. Usually the ramification index is one. But if the ramification index is not equal to one, then P is by definition a ramification point, and Q is a branch point.

If Y is just the Riemann sphere, and Q is in the finite part of Y, then there is no need to select special coordinates. The ramification index can be calculated explicitly from Cauchy's integral formula. Let γ be a simple rectifiable loop in X around P. The ramification index of ƒ at P is

This integral is the number of times ƒ(γ) winds around the point Q. As above, P is a ramification point and Q is a branch point if e_P > 1.