Editing Monodromy (section)

==Example==
These ideas were first made explicit in [[complex analysis]]. In the process of [[analytic continuation]], a function that is an [[analytic function]] <math>F(z)</math> in some open subset <math>E</math> of the punctured complex plane <math>\mathbb{C} \backslash \{0\}</math> may be continued back into <math>E</math>, but with different values. For example, take

: <math>\begin{align}
  F(z) &= \log(z) \\
     E &= \{z\in \mathbb{C} \mid \operatorname{Re}(z)>0\}.
\end{align}</math>

Then analytic continuation anti-clockwise round the circle

: <math>|z| = 1</math>

will result in the return not to <math>F(z)</math> but to

: <math>F(z) + 2\pi i.</math>

In this case the monodromy group is the [[infinite cyclic group]], and the covering space is the universal cover of the punctured [[complex plane]]. This cover can be visualized as the [[helicoid]] with [[parametric equation]]s <math>(x, y, z) = (\rho \cos \theta, \rho \sin \theta, \theta)</math> restricted to <math>\rho > 0</math>. The covering map is a vertical projection, in a sense collapsing the spiral in the obvious way to get a punctured plane.