# 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

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

How to Cite This Entry:
Winding number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Winding_number&oldid=49226