Why does $z^{-1}$ not have an antiderivative? I had been given the question as shown in the following, with the answer also given.

EXAMPLE 6.13. Let $\gamma$ be the circle with centre 0 and radius 1 traced anticlockwise. By integrating along $\gamma$, show that there is no function on $\mathbb{C} \backslash\{0\}$ with derivative $z^{-1}$.
SOLUTION. We can parametrise $\gamma$ as $z=e^{i t}$ for $0 \leq t \leq 2 \pi$. We have
$$
d z=i e^{i t} d t=i z d t
$$
hence
$$
\int_{\gamma} z^{-1} d z=\int_{0}^{2 \pi} z^{-1} i z d t=\int_{0}^{2 \pi} i d t=2 \pi i \neq 0
$$
Since $\gamma$ is a closed path, it follows that $z^{-1}$ cannot have an anti-derivative on $\mathbb{C} \backslash\{0\}$.

Surely however the anti-derivative of $z^{-1}$ would be $\log(z)$? I have seen a similar question asked, however, I didn't see the answer to this specific question.
 A: $\DeclareMathOperator{\Re}{Re}\DeclareMathOperator{\Im}{Im}$Because the integral of the reciprocal about the unit circle is non-zero, the value of the integral
$$
w = \int_{1}^{z} \frac{d\zeta}{\zeta}
\tag{1}
$$
depends not only on $z$, but on the path from $1$ to $z$. You can denote the value of the integral $w = \log z$, but doing so does not make $\log$ single-valued. The green paths below depict two "lifts" of the unit circle, starting at $(1, 0)$ and tracing counterclockwise, ending at $(1, 2\pi i)$; tracing clockwise, ending at $(1, -2\pi i)$.
For each non-zero $z$, there are infinitely many distinct values of $\log z$. Fixing a path $\gamma$ from $1$ to $z$ fixes a value of
$$
\log z = \int_{\gamma} \frac{d\zeta}{\zeta}.
$$
Every other choice differs by an added integer multiple of $2\pi i$, corresponding to a path that winds about the origin some number of times before following $\gamma$ to $z$.
There's no perfectly adequate picture because the graph of the relation $w = \log z$ naturally sits in the space of ordered pairs $(z, w)$ of complex numbers (a real four-dimensional space), but the projection
$$
(z, w) \mapsto (\Re z, \Im z, \Re w + \Im w)
$$
(which perhaps is not as widely-known as it should be), captures the qualitative "parking garage" behavior of the Riemann surface of $\log$.

