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Cauchy-Goursat: If $f$ is analytic in a simply connected domain D, then $\int_{c}f(z) dz=0 $ for any closed curve $c$ in D.

However, I have seen questions where Cauchy-Goursat has been used when the function is just analytic in and on a closed curve C. Where are the simply connected regions in these situations? I know that Cauchy Gourmet can be applied to curves within C but why is it possible on C itself?

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  • $\begingroup$ The key is any closed curve. $\endgroup$ – Jacky Chong Apr 2 '18 at 3:35
  • $\begingroup$ Yes but it is a closed curve in D. How about the boundary of D? Is C itself in any simply connected region? If so it that region determined by the open balls wise centers are points n the curve? $\endgroup$ – Jhon Doe Apr 2 '18 at 3:39

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