The winding number is the circulation of the planar vortex field divided by $2\pi$

The hypothesis of this remark is piecewise $C^1$, not merely rectifiable. The real line integrals it quotes are those of Scalar line integrals with respect to arc length and vector-field line integrals, which are defined through a derivative of the path; a general complex contour is only rectifiable and has no derivative, so the identification below is asserted only for a piecewise-$C^1$ closed contour $\gamma :[a,b]\to \mathbb{C}\setminus \{0\}$. Read the plane as ${\mathbb{R}}^2$ through $\mathbb{C}=\mathbb{R}[x]\mathrm{/}(x^2+1)$ as the Euclidean plane and as a normed real algebra: what the identification preserves and write $\gamma =(x,y)$ for the corresponding planar path.

The split. With $\mathrm{Re}$, $\mathrm{Im}$ and the modulus as in Real and imaginary parts, complex conjugation, and modulus, the identity $1\mathrm{/}z=\bar{z}\mathrm{/}\mathrm{\mid }z{\mathrm{\mid }}^2$ turns the integrand into

$\frac{dz}{z}=\frac{(x-iy)(dx+idy)}{x^2+y^2}=\frac{xdx+ydy}{x^2+y^2}+i\frac{xdy-ydx}{x^2+y^2},$

and For piecewise-C1 contours the Riemann–Stieltjes integral agrees with the parametric complex integral and the published real line integrals identifies the two real parts of ${\int }_{\gamma }dz\mathrm{/}z$ with the vector line integrals of

$P(x,y)=(\frac{x}{x^2+y^2},\frac{y}{x^2+y^2}),V(x,y)=(\frac{-y}{x^2+y^2},\frac{x}{x^2+y^2})$

along $\gamma$.

The first field contributes nothing. $P$ is the gradient of $\varphi (x,y)=\frac{1}{2}\log (x^2+y^2)$ on the punctured plane, so The gradient theorem: the line integral of a gradient is the endpoint increment evaluates its line integral as $\varphi (\gamma (b))-\varphi (\gamma (a))$, which is $0$ for a closed path; the same conclusion is what Conservative fields are path-independent and have zero integral around every closed path records for a conservative field.

The second field is the vortex field. $V$ is the $C^1$ field on ${\mathbb{R}}^2\setminus \{(0,0)\}$ whose partial derivatives satisfy the closedness condition of Exact and closed C1 vector fields while admitting no global potential there. Since ${\int }_{\gamma }\frac{dz}{z}=2\pi in(\gamma ,0)$ by The winding number of a closed contour about a point off its trace, the circulation of $V$ around $\gamma$ is $2\pi n(\gamma ,0)$.

What the two statements share. A nonzero winding number and the failure of $V$ to be exact on the punctured plane are the same fact recorded in two vocabularies: if $V$ had a $C^2$ potential on the punctured plane, then The gradient theorem: the line integral of a gradient is the endpoint increment would force its circulation, and hence $n(\gamma ,0)$, to vanish around every closed piecewise-$C^1$ path there; A circle traversed $k$ times has winding number $k$ inside and $0$ outside exhibits circles with index $k$ about the origin for every integer $k$. That exactness needs more than closedness on a domain of this shape is the point recorded in Closedness is local, exactness is global, and a domain hypothesis cannot be omitted, and Every exact C1 vector field is closed is the implication that does hold on every open set.