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In complex analysis, the proof of Morera's Theorem

"for $f\in C(D) $ such that D is a region, if for any triangle $\triangle$ in D, $\int_{\triangle}f = 0$ is True, then f is analytic in D."

splits the proof into two cases: for convex and non-convex regions D.

I'd like some intuition before my exam on what a non-convex region might look like. Can you provide an example?


EDIT:

I apologize gentlemen, this question has shown me that the definition of a region is regrettably localized by textbook.

In Bak-Newman, a region is defined as an open, connected set in $\mathbb C$, which would make @Did's examples valid if considered with their boundaries removed in $\mathbb C$.

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  • $\begingroup$ Did's examples are invalid? Can you not imagine a banana-shaped open connected set in $\mathbb C$? $\endgroup$ – user856 Oct 18 '13 at 0:59
  • $\begingroup$ Yes, I understand how to extend his examples to satisfy my definition. I'd like him to change his answer to suit my edit so I can accept it. $\endgroup$ – Jake Color Oct 18 '13 at 1:03
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A banana. A necklace. A tire. An eight (actually no Roman digit is convex except "one" when it is written as a vertical bar instead of as $1$).

Take the drawing of the digit 7 for example. The segment joining its two ends is not included in the drawing. If the figure 7 was convex, it would contain this whole segment.

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  • $\begingroup$ "Tyre" = "tire"? $\endgroup$ – Pete L. Clark Oct 17 '13 at 6:21
  • $\begingroup$ Yes... en.wikipedia.org/wiki/Tire $\endgroup$ – Did Oct 17 '13 at 6:22
  • $\begingroup$ @Did please see my edit.. I took for granted the universality of the definition of a region. $\endgroup$ – Jake Color Oct 18 '13 at 0:56
  • $\begingroup$ Sorry but contrarily to what you write in the added part of your post, one does not have to do anything to make these examples valid, not even to extend them. They are valid. $\endgroup$ – Did Oct 18 '13 at 5:31
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Something with a hole in the middle is non-convex, but can still be connected. In particular, $\mathbb{C} \setminus \{0\}$.

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A circle in the plane is , maybe, as non-convex as you can get, in that there are no two points in the circle that can be joined by a line that lays inside of the circle. But a closed or open disk is convex.

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