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

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

for $\varepsilon>0$ sufficiently small, such that we enclose only one root of $f$.

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 perhaps a well-known construction?

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

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

for $\varepsilon>0$ sufficiently small, such that we enclose only one root of $f$.

I am trying to understand if this defines a reasonable winding number in the sense that it is discrete-valued $\in 2\pi \mathbb Z$. I don't quite see why this is the case. Is this perhaps a well-known construction?

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

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

for $\varepsilon>0$ sufficiently small, such that we enclose only one root of $f$.

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 perhaps a well-known construction?

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

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

for $\varepsilon>0$ sufficiently small, such that we enclose only one root of $f$.

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 perhaps a well-known construction?

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

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

for $\varepsilon>0$ sufficiently small, such that we enclose only one root of $f$.

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 perhaps a well-known construction?

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

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

for $\varepsilon>0$ sufficiently small, such that we enclose only one root of $f$.

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 perhaps a well-known construction?