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I'm reading Conway's complex analysis book and on page 84 he says:

Quite often I see this version of the theorem:

Let $f$ be analytic in the simply connected domain $D$ and let $C$ be a simple closed positively oriented contour that lies in $D$. If $z_0$ is a point that lies interior to $C$, then $$f(z_0)=\frac{1}{2\pi i}\int_C\frac{f(z)}{z-z_0}$$ (From this book page 235)

The Conway's formulation seems more general. The problem is I can't understand why Conway's formulation implies in the second formulation. Is the winding number in the case of the second formulation one?

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    $\begingroup$ That the winding number is one, is exactly en.wikipedia.org/wiki/Jordan_curve_theorem (as $z_0$ is in the interior). And this is the reason why the first theorem implies the second $\endgroup$ – b00n heT Aug 29 '16 at 13:50
  • $\begingroup$ @b00nheT but the winding number is zero when $\gamma$ is homotopic to a constant, no? $\endgroup$ – user42912 Aug 29 '16 at 19:02
  • $\begingroup$ One can derive the first theorem without the knowledge of Jordan curve theorem & the theorem that says that the winding number around Jordan curve is $\pm 1$. But note: there is a different definition for "simply connected domain" that considers homotopic curves and not Jordan curves, i.e. simply connected domain is a region where we can deform any loop into a point. But if you know the Jordan curve theorem, then you can immediately define simple connected domain without any hocus pocus... $\endgroup$ – Hulkster Aug 29 '16 at 19:38
  • $\begingroup$ @Juho Please could you explain the relationship between my doubt and the Jordan Curve theorem. I didn't understand what this theorem has to do with everything else. Thank you $\endgroup$ – user42912 Aug 29 '16 at 22:28
  • $\begingroup$ Note also that you can't always deform loop $\gamma$ into a point without crossing the point $a$. $\endgroup$ – Hulkster Aug 29 '16 at 23:33
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Topology - Munkres 2ed, p. 403

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The winding number may in fact be 'defined' as $$ n(\gamma;a)= \frac{1}{2\pi i} \oint_\gamma \frac{1}{z-a} \; dz$$ and this is typically also used in the proof. In Conwey's version $G$ need not be simply connected, $\gamma$ may intersect itself, but $\gamma$ should not effectively 'wind around' any points in the complement of $G$. The winding number is indeed one in the second formulation.

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  • $\begingroup$ But if $\gamma$ doesn't "wind around" then the winding number is zero, no? $\endgroup$ – user42912 Aug 29 '16 at 14:04
  • $\begingroup$ Indeed. When $\gamma$ is the boundary of an open domain $U$ and goes around counter-clockwise (this corresponds to a curve $C$ as in the second formulation) then $n(\gamma;a)=+1$ if $a\in U$ and zero otherwise. Conwey's formulation allows for the curve $\gamma$ to wind around and then unwind again. For example, if you have an annulus the geometric image of $\gamma$ may in fact encircle the hole in the middle! but still have winding number 0 around points in the hole (you should probably make a drawing for this). $\endgroup$ – H. H. Rugh Aug 29 '16 at 14:12

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