Winding number - Knowlpedia

Definition

Let $\gamma$ be a closed piecewise $C^1$ contour in $\mathbb C$ , and let $a\notin\gamma$ . Its winding number or index about $a$ is

\operatorname{Ind}(\gamma,a) =\frac{1}{2\pi i}\int_\gamma\frac{dz}{z-a}.

This number is an integer and is constant as $a$ varies within a connected component of $\mathbb C\setminus\gamma$ .

Interpretation

Choose a continuous argument of $\gamma(t)-a$ locally along the contour. Its total change after one circuit is $2\pi\operatorname{Ind}(\gamma,a)$ . Positive counterclockwise circuits contribute $+1$ ; reversing orientation negates the index.

Homotopy behavior

The winding number is unchanged under a homotopy of closed contours that avoids $a$ . It therefore records the class of the contour in the punctured plane and supplies the coefficients in the general Cauchy integral formula and residue theorem.

References

John B. Conway, Functions of One Complex Variable I, 2nd ed., Springer, 1978. Publisher record. Relevant: Chapter IV, §§2–4.