Let $f:\mathbb R^2 \to \mathbb C$ be a $C^1$ function that vanishes at a point $x_0.$

I can then define $\lim_{\varepsilon \downarrow 0}-i \int_{\gamma_{\varepsilon}} \nabla \log(f(s)) \cdot ds = - i \lim_{\varepsilon \downarrow 0} \int_0^1 \nabla (\log f)(\gamma_{\varepsilon}(t)) \cdot \gamma_{\varepsilon}'(t) \ dt,$ where $$\gamma_{\varepsilon}(t) = x_0 + \varepsilon (\cos(2\pi t),\sin(2\pi t)).$$

I am trying to understand if this defines a reasonable winding number in the sense that it is discrete-valued. I don't quite see why this is the case. Is this somehow a well-known construction? - Do I need the limit $\varepsilon \downarrow 0$ or is it independent of this parameter, as long as I only enclose one zero of $f$?