Let $\gamma$ denote the unit circle parameterized on the domain $[0,2\pi]$.
I'm trying to compute $n(\gamma, 0)$ as follows:
$$ n(\gamma,0) = {1 \over 2\pi i}\int_\gamma {dz \over z} = {1 \over 2 \pi i} \int_0^{2 \pi} {1 \over \gamma(t)} \gamma'(t)\ dt= \underbrace{\ldots}_{\text{?}} = 1 $$
As flagged above, I'm not sure what justifies the inference to the answer of $1$.
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Because $\gamma(t)=e^{it},\;t\in[0,2\pi]$. Hence $$ \frac{1}{2\pi i}\int_0^{2\pi}\frac{\gamma'(t)}{\gamma(t)}\,dt =\frac{1}{2\pi i}\int_0^{2\pi}\frac{ie^{it}}{e^{it}}\,dt=1$$
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$\begingroup$ I don't think he should/must/would, @Joe. Whoever's doing complex analysis but cannot understand the parametrization of the unit circle should, perhaps, go back to the basics in geometry. Nice answer. +1 $\endgroup$ – DonAntonio Apr 18 '14 at 22:00
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