# 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 |

**How to Cite This Entry:**

Winding number.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Winding_number&oldid=51094