Editing Branch point (section)

==Algebraic branch points==

Let <math>\Omega</math> be a connected [[open set]] in the [[complex plane]] <math>\mathbb{C}</math> and <math>f:\Omega\to\mathbb{C}</math> a [[holomorphic function]].  If <math>f 
</math> is not constant, then the set of the [[critical point (mathematics)|critical point]]s of <math>f 
</math>, that is, the zeros of the derivative <math>f'(z)</math>, has no [[limit point]] in <math>\Omega</math>.  So each critical point <math>z_0</math> of <math>f 
</math> lies at the center of a disc <math>B(z_0,r)</math> containing no other critical point of <math>f 
</math> in its closure.

Let <math>\gamma</math> be the boundary of <math>B(z_0,r)</math>, taken with its positive orientation.  The [[winding number]] of <math>f(\gamma)</math> with respect to the point <math>f(z_0)</math> is a positive integer called the '''[[Ramification (mathematics)|ramification]] index''' of <math>z_0</math>.  If the ramification index is greater than 1, then <math>z_0</math> is called a '''ramification point''' of <math>f 
</math>, and the corresponding [[critical value (critical point)|critical value]] <math>f(z_0)</math> is called an (algebraic) '''branch point'''.  Equivalently, <math>z_0</math> is a ramification point if there exists a holomorphic function <math>\phi
</math>  defined in a neighborhood of <math>z_0</math> such that <math>f(z) = \phi(z)(z-z_0)^k + f(z_0)
</math> for integer <math>k>1</math>.

Typically, one is not interested in <math>f 
</math> itself, but in its [[inverse function]].  However, the inverse of a holomorphic function in the neighborhood of a ramification point does not properly exist, and so one is forced to define it in a multiple-valued sense as a [[global analytic function]].  It is common to [[abuse of terminology|abuse language]] and refer to a branch point <math>w_0 = f(z_0)
</math> of <math>f 
</math> as a branch point of the global analytic function <math>f^{-1}
</math>.  More general definitions of branch points are possible for other kinds of multiple-valued global analytic functions, such as those that are defined [[implicit function|implicitly]].  A unifying framework for dealing with such examples is supplied in the language of [[Riemann surface]]s below.  In particular, in this more general picture, [[pole (complex analysis)|poles]] of order greater than 1 can also be considered ramification points.

In terms of the inverse global analytic function <math>f^{-1}
</math>, branch points are those points around which there is nontrivial monodromy.  For example, the function <math>f(z) = z^2
</math> has a ramification point at <math>z_0 = 0
</math>.  The inverse function is the square root <math>f^{-1}(w) = w^{1/2}
</math>, which has a branch point at <math>w_0 = 0
</math>.  Indeed, going around the closed loop <math>w = e^{i\theta}</math>, one starts at  <math>\theta = 0</math> and <math>e^{i0/2} = 1
</math>.  But after going around the loop to <math>\theta = 2\pi
</math>, one has <math>e^{2\pi i/2} = -1
</math>. Thus there is monodromy around this loop enclosing the origin.