I have read a book about Complex Variable and I arose a doubt: Let $f$ be a continuous function on an open ($\Omega$) of C. Suppose that $f$ verifies the thesis of Cauchy-Goursat theorem for any triangle whose edges are contained in $\Omega$.

Is $f$ holomorphic in $\Omega$? Does primitive $f$?

I don't know if I explain well. Thank you,beforehand.


This is known as Morera's theorem, and it shows that f is holomorphic.

Edited in response to comments.

  • 2
    $\begingroup$ actually, they have primitives on simply connected domains. the function 1/z is holomorphic on $\mathbb{C}-\{0\}$ but it has no primitive on $\mathbb{C}-\{0\}$ $\endgroup$ – Glougloubarbaki Feb 3 '13 at 15:24
  • $\begingroup$ Does it have primitives? $\endgroup$ – Sophie Germain Feb 3 '13 at 15:59

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