Winding number

Let $ \Gamma = \{ {z( \tau ) } : {\alpha \leq \tau \leq \beta } \} $ be an arc in the complex plane and let $ c $ be a point not on $ \Gamma $. A continuous argument of $ z- c $ on $ \Gamma $ is a continuous real-valued function $ \phi $ on $ [ \alpha , \beta ] $ that for each $ \tau \in [ \alpha , \beta ] $ is an argument of $ z ( \tau ) - c $, i.e. $ z ( \tau ) - c = r \mathop{\rm exp} ( i \phi ( \tau )) $ for some $ r $. Such functions can be found, and if $ \phi ( \tau ) $, $ \psi ( \tau ) $ are two continuous arguments, then they differ by a constant integral multiple of $ 2 \pi $. It follows that the increase of the argument, $ \phi ( \beta ) - \phi ( \alpha ) $, does not depend on the choice of the continuous argument. It is denoted by $ [ \mathop{\rm arg} z ( \tau ) - c ] _ \Gamma $. If $ \Gamma $ is a piecewise-regular arc,

$$ [ \mathop{\rm arg} z ( \tau ) - c ] _ \Gamma = \ \mathop{\rm Im} \int\limits _ \Gamma \frac{1}{z-c} dz . $$

In the special case that $ \Gamma $ is a closed curve, i.e. $ z ( \alpha ) = z ( \beta ) $, $ [ \mathop{\rm arg} z ( \tau ) - c ] _ \Gamma $ is necessarily an integral multiple of $ 2 \pi $ and the integer

$$ n ( \Gamma , c ) = \frac{1}{2 \pi } [ \mathop{\rm arg} z( \tau ) - c] _ \Gamma $$

is called the winding number of $ \Gamma $ with respect to $ c $. For a piecewise-regular closed curve $ \Gamma $ with $ c $ not on $ \Gamma $ one has

$$ n ( \Gamma , c ) = \frac{1}{2 \pi i } \int\limits _ \Gamma \frac{1}{z-c} dz . $$

References

[a1]	P. Henrici, "Applied and computational complex analysis" , 1 , Wiley (Interscience) (1974) pp. §4.6