I've noticed that the definition of a winding number is rather different in Stewart & Tall's Complex Analysis than in Ahlfors' Complex Analysis:
Let $\gamma:[a,b]\to\mathbb{C}$ be an arc and let $a\in\mathbb{C}$ lie outside its image.
Definition (Stewart, Tall): the winding number $w(\gamma,a)$ is defined as $$\frac{\theta(b)-\theta(a)}{2\pi}$$ for any continuous choice of argument along $\gamma$ i.e. for any continuous map $\theta:[a,b]\to\mathbb{R}$ such that $$e^{i\theta(t)} = \frac{\gamma(t)}{|\gamma(t)|}.$$
Definition (Ahlfors): the winding number $n(\gamma,a)$ is defined as $$\frac{1}{2\pi i}\int_\gamma\frac{1}{z-a}.$$
How can one show these two definitions are equivalent?
I shall assume that $\gamma$ is a closed curve.
Let $[b,c]$ be the domain of $\gamma$ and define$$\begin{array}{rccc}F\colon&[b,c]&\longrightarrow&\Bbb C\\&t&\mapsto&\displaystyle\int_b^t\frac{\gamma'(u)}{\gamma(u)-a}\,\mathrm du.\end{array}$$I shall prove that$$(\forall t\in[b,c]):\gamma(t)-a=(\gamma(b)-a)e^{F(t)}.\label{a}\tag1$$Assertion \eqref{a} is clearly true when $t=b$. In order to prove that it is true in general, it is enough to prove that $(\gamma-a)e^{-F}$ is constant, which is true, since you get $0$ when you differentiate it (note that this uses the fact, that follows from the definition of $F$, that $F'=\frac{\gamma'}{\gamma-a}$). It follows that\begin{align}\gamma(b)-a&=\gamma(c)-a\text{ (since $\gamma$ is a closed curve)}\\&=(\gamma(b)-a)e^{F(c)},\end{align}and therefore $e^{F(c)}=1$. So, $F(c)\in2\pi i\Bbb Z$.
Now, take $\alpha\in\Bbb R$ such that $\alpha$ is an argument of $\gamma(b)-a$. It follows from \eqref{a} that, for each $t\in[b,c]$,\begin{align}\gamma(t)-a&=|\gamma(b)-a|e^{i\alpha}e^{F(t)}\\&=|\gamma(b)-a|e^{\operatorname{Re}f(t)}e^{i(\alpha+\operatorname{Im}F(t))}\end{align}and therefore the map$$\begin{array}{rccc}\theta^*\colon&[b,c]&\longrightarrow&\Bbb R\\&t&\mapsto&\alpha+\operatorname{Im}F(t)\end{array}$$is a continuous map and, for each $t\in[a,b]$, $\theta^*(t)$ is an argument of $\gamma(t)-a$. And\begin{align}\frac1{2\pi i}\int_\gamma\frac1{z-a}\,\mathrm dz&=\frac{F(c)-F(b)}{2\pi i}\\&=\frac{F(c)}{2\pi i}\\&=\frac{\operatorname{Im}F(c)}{2\pi}\text{ (since $F(c)\in2\pi i\Bbb Z$)}\\&=\frac{\theta^*(c)-\theta^*(b)}{2\pi}.\end{align}And now, if $\theta\colon[b,c]\longrightarrow\Bbb R$ is an arbitrary continuous functions such that, for each $t\in[b,c]$, $\theta(t)$ is an argument of $\gamma(t)-a$, then it is easy to prove that $\theta$ and $\theta^*$ differ by a constant, and therefore\begin{align}\frac{\theta(c)-\theta(b)}{2\pi}&=\frac{\theta^*(c)-\theta^*(b)}{2\pi}\\&=\frac1{2\pi i}\int_\gamma\frac1{z-a}\,\mathrm dz.\end{align}