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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
:<math>e_P = \frac{1}{2\pi i}\int_\gamma \frac{f'(z)}{f(z)-f(P)}\,dz.</math>
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''<sub>''P''</sub> > 1.
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