Editing Winding number (section)

==Formal definition==

Let <math>\gamma:[0,1] \to \Complex \setminus \{a\}</math> be a continuous closed path on the plane minus one point. The winding number of <math>\gamma</math> around <math>a</math> is the integer

:<math>\text{wind}(\gamma,a) = s(1) - s(0),</math>

where <math>(\rho,s)</math> is the path written in polar coordinates, i.e. the lifted path through the [[covering space|covering map]]

:<math>p:\Reals_{>0} \times \Reals \to \Complex \setminus \{a\}: (\rho_0,s_0) \mapsto a+\rho_0 e^{i2\pi s_0}.</math>

The winding number is well defined because of the [[Covering space#Lifting property|existence and uniqueness of the lifted path]] (given the starting point in the covering space) and because all the fibers of <math>p</math> are of the form <math>\rho_0 \times (s_0 + \Z)</math> (so the above expression does not depend on the choice of the starting point). It is an integer because the path is closed.