Cauchy's Integral Theorem says that if a function is analytic in open and simply connected domain and $\gamma$ is a closed curve so:


Morera Theorem says that if a function is continuous on an open domain such that for every closed curve $$\int_{\gamma}f(z)dz=0$$ So: the function is analytic

So those both theorem are 2 different directions of iff statement?

  • $\begingroup$ What's the question? $\endgroup$ – user228113 Jun 16 '17 at 9:16
  • $\begingroup$ You can then define $F(z)$ as the integral of $f(z)$ from some fixed $z_0$ to $z$ along an arbitrary path, the way that path is chosen doesn't matter as the contour integral is zero. You can then show that $F(z)$ is differentiable, the derivative is $f(z)$ so $F(z)$ is analytic, which makes $f(z)$ analytic too. $\endgroup$ – Count Iblis Jun 16 '17 at 9:18
  • $\begingroup$ @G.Sassatelli Sorry, added the question $\endgroup$ – gbox Jun 16 '17 at 9:22

Morera Theorem is usually considered a converse of the Cauchy integral theorem but it is not (usually) presented as an iff statement because the two theorems can be formulated with some differences on the conditions.

Using the same conditions we can say that, if $f$ is a continuous function on simply-connected region $D$, the the Cauchy theorem says:

$f$ is analytic $\Rightarrow$ $f$ has an antiderivative

and the Morera's theorem says

$f$ has an antiderivative $\Rightarrow$ $f$ is analytic


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