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I have a question about this statement in the Cauchy's Integral Formula in Conway text.
In the Integral formula, it states that

" Let $G$ be an open subset of the plane... If $\gamma$ is a closed rectifiable curve in $G$ such that $n(\gamma;w) = 0 $ for all $w \in \mathbb{C}-G $."

Now before this theorem, Conway presents a theorem about winding number, it states that "Let $ \gamma$ be a closed rectifiable curve in $\mathbb{C}.$ Then $n(\gamma;a)$ is constant for $a$ belonging to a component of $B= \mathbb{C}-\{\gamma\} $. Also, $ n(\gamma;a) = 0 $ for $a$ belonging to the unbounded component of $B$. "

My question is, why does the statement of the Integral Formula have to say ".. such that $n(\gamma;w) = 0 $ for all $w \in \mathbb{C}-G $." ??

Because by the theorem about winding number above, since $\gamma \in G$, don't we already know that $n(\gamma;w) = 0 $ for all $w \in \mathbb{C}-G $??

sorry for the confusing question, thank you!!

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If the open subset $G$ were an annulus, then $\mathbb{C}\setminus G$ has two connected components. Suppose $\gamma$ winds around the hole in the annulus. In the unbounded component of $\mathbb{C}\setminus G$ for sure $n(\gamma;w)=0$ since it's part of the unbounded component of $\mathbb{C}\setminus \gamma$, but that's not necessarily true in the bounded component, per the results you've given. (An annulus is for example $\{z:1<|z|<2\}$.)

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  • $\begingroup$ Oh, I see now, thank you for your help $\endgroup$ – Khoa ta Feb 5 '16 at 2:57

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