[proofplan] We use the polar lifting of the closed path $\gamma$ about $w$: write $\gamma(t) - w = r(t)e^{i\theta(t)}$ with $r, \theta$ continuous. Computing $\gamma'(t)/(\gamma(t) - w)$ in polar form and integrating gives a real part that vanishes (since $\gamma$ is closed, $r(a) = r(b)$) and an imaginary part equal to $\theta(b) - \theta(a) = 2\pi \, I(\gamma, w)$. [/proofplan] [step:Write $\gamma(t) - w$ in polar form using the lifting lemma] Since $w \notin \gamma^*$, the function $t \mapsto \gamma(t) - w$ is a continuous non-vanishing curve in $\mathbb{C} \setminus \{0\}$. By the lifting lemma, there exist continuous functions $r: [a,b] \to (0, \infty)$ and $\theta: [a,b] \to \mathbb{R}$ such that \begin{align*} \gamma(t) - w = r(t) \, e^{i\theta(t)} \quad \text{for all } t \in [a,b]. \end{align*} Differentiating: $\gamma'(t) = (\dot{r}(t) + i r(t) \dot{\theta}(t)) \, e^{i\theta(t)}$. [/step] [step:Compute $\gamma'/(\gamma - w)$ and integrate to obtain the winding number] Dividing by $\gamma(t) - w = r(t) e^{i\theta(t)}$: \begin{align*} \frac{\gamma'(t)}{\gamma(t) - w} = \frac{\dot{r}(t) + i r(t) \dot{\theta}(t)}{r(t)} = \frac{\dot{r}(t)}{r(t)} + i \dot{\theta}(t). \end{align*} Integrating over $[a,b]$: \begin{align*} \int_\gamma \frac{dz}{z - w} = \int_a^b \frac{\gamma'(t)}{\gamma(t) - w} \, dt = \int_a^b \left(\frac{\dot{r}(t)}{r(t)} + i \dot{\theta}(t)\right) dt = \bigl[\ln r(t) + i\theta(t)\bigr]_a^b. \end{align*} Since $\gamma$ is closed, $\gamma(a) = \gamma(b)$, so $r(a) e^{i\theta(a)} = r(b) e^{i\theta(b)}$. Taking absolute values: $r(a) = r(b)$, hence $\ln r(b) - \ln r(a) = 0$. Therefore \begin{align*} \frac{1}{2\pi i} \int_\gamma \frac{dz}{z - w} = \frac{1}{2\pi i} \cdot i(\theta(b) - \theta(a)) = \frac{\theta(b) - \theta(a)}{2\pi} = I(\gamma, w), \end{align*} since the winding number $I(\gamma, w)$ is defined as the total change in argument divided by $2\pi$: $I(\gamma, w) = (\theta(b) - \theta(a))/(2\pi)$. [guided] The real part $\int_a^b \dot{r}/r \, dt = \ln r(b) - \ln r(a)$ measures the net change in $\ln|\gamma(t) - w|$. For a closed path, $|\gamma(b) - w| = |\gamma(a) - w|$, so this contribution vanishes. The imaginary part $\int_a^b \dot{\theta} \, dt = \theta(b) - \theta(a)$ measures the total change in the argument of $\gamma(t) - w$ as $t$ traverses $[a,b]$. This is precisely $2\pi$ times the winding number: $I(\gamma, w)$ counts how many times $\gamma$ winds around $w$, and each complete counter-clockwise loop adds $2\pi$ to $\theta$. The lifting lemma guarantees that a continuous branch of $\theta(t)$ exists even though $\arg$ is multi-valued. The key hypothesis is that $\gamma(t) - w \neq 0$ for all $t$, so the curve lives in $\mathbb{C} \setminus \{0\}$, which has the real line as a universal cover via $\theta \mapsto e^{i\theta}$. [/guided] [/step]