Goal: Let $\gamma$ be the unit circle. Then I aim to compute
$$ \int_{|z|=1} {e^z \over z}\ dz = \int_{\gamma} {e^z \over z}\ dz $$
Attempt:
Consider that $\gamma$ is a closed curve.
Let $a = 0$. Then $e^a = 1$. Furthermore, $\gamma \cap \{a\} = \emptyset$.
We have by another theorem if $f$ is analytic inside some disk $\Delta$, and if $\gamma$ is a closed curve in $\Delta$ that does not intersect some point $a \in \Delta$, then
$$ f(a) \cdot n(\gamma, a) = {1 \over 2 \pi i} \int_\gamma {f(z) \over z-a}\ dz $$
Then combining the facts from above with the fact that $e^z$ is analytic on the whole plane (hence analytic inside any disk that contains the unit circle), we have that
$$ \int_\gamma {e^z \over z - a}\ dz = f(a) n(\gamma, a) = n(\gamma, 0) $$
Question: Is my reasoning correct? Is there an easy way we compute the exact integer value of $n(\gamma,0)$?
