Winding number

Given a closed contour $γ : [0, 1] \to C ∖ a$ , the winding number $n (a, γ)$ measures how many times $γ$ loops around the point $a$ , counting orientation. Formally,

n (a, γ) = \frac{1}{2 π i} \int_{γ} \frac{1}{z - a}, d z .

It is an integer invariant under any homotopy of $γ$ that avoids $a$ . When $n (a, γ) = 1$ , the contour encircles $a$ once counterclockwise; when $0$ , it doesn’t encircle it at all. This single integer is the topological ingredient behind Cauchy integral formula.