A circle traversed $k$ times has winding number $k$ inside and $0$ outside

By [L8] one has ${\gamma }_k(t)=a+r\cos (kt)+ir\sin (kt)$, which by [L11] is differentiable in $t$ with the continuous derivative $-kr\sin (kt)+ikr\cos (kt)$, so ${\gamma }_k$ is rectifiable by [L12]; and ${\gamma }_k(2\pi )=a+r\exp (2\pi ik)=a+r={\gamma }_k(0)$ by [L9], so ${\gamma }_k$ is a closed complex contour.

Since $\mathrm{\mid }{\gamma }_k(t)-a\mathrm{\mid }=r\mathrm{\mid }\exp (ikt)\mathrm{\mid }=r>0$ by [L8] and [L17], the centre $a$ lies off the trace, and $\lambda (t)=\log r+ikt$ is a continuous map on $[0,2\pi ]$ with $\exp (\lambda (t))=e^{\log r}\exp (ikt)=r\exp (ikt)={\gamma }_k(t)-a$ by [L8] and [L13]; so $\lambda$ is a continuous logarithm of ${\gamma }_k-a$ along ${\gamma }_k$ in the sense of [L6].

The trace of ${\gamma }_k$ is $\{a+r\exp (ikt):0\le t\le 2\pi \}$. If $k=0$ this is $\{a+r\}$. If $k\mathrm{\ne }0$ then $\{kt:0\le t\le 2\pi \}$ is a closed interval of length $2\pi \mathrm{\mid }k\mathrm{\mid }\ge 2\pi$, so by the periodicity and surjectivity in [L10] the values $\exp (iks)$ run over the whole unit circle, and the trace is $\{z:\mathrm{\mid }z-a\mathrm{\mid }=r\}$ by [L8] and [L17]. In both cases the trace is contained in $\{z:\mathrm{\mid }z-a\mathrm{\mid }=r\}$.

By [L6] and step 1.2, ${\int }_{{\gamma }_k}dz\mathrm{/}(z-a)=\lambda (2\pi )-\lambda (0)=2\pi ik$, so $n({\gamma }_k,a)=k$ by [L1]; for $k=1$ this is the published normalisation [L7], and by [L2] and [L18] the value is an integer, as it must be.

The disc $D=B(a,r)$ is convex by [L14] and [L17], hence path-connected along segments and therefore connected; step 1.3 puts the trace in $\{z:\mathrm{\mid }z-a\mathrm{\mid }=r\}$, which is disjoint from $D$, so $D\subseteq \mathbb{C}\setminus {\gamma }_k^{\ast }$ and $a\in D$.

The set $E=\{z:\mathrm{\mid }z-a\mathrm{\mid }>r\}$ is connected by [L5] and is disjoint from the trace by step 1.3; it is unbounded by [L16] and [L17], so by [L15] it lies in a single component of $\mathbb{C}\setminus {\gamma }_k^{\ast }$, and that component is unbounded, hence is the unique unbounded one of [L4].

By [L3] the index is constant on the component of $\mathbb{C}\setminus {\gamma }_k^{\ast }$ containing $D$, and $D$ is a connected subset of that complement containing $a$, so by [L15] it lies in one component; hence $n({\gamma }_k,z)=n({\gamma }_k,a)=k$ for every $z\in D$, that is for $\mathrm{\mid }z-a\mathrm{\mid }<r$.

By [L4] the index vanishes on that unique unbounded component, so $n({\gamma }_k,z)=0$ for every $z\in E$ by step 2.3, while step 3.1 gives the value $k$ on $\mathrm{\mid }z-a\mathrm{\mid }<r$; when $k=0$ both regions still lie off the single-point trace of step 1.3 and both values are $0$.