# Morera Theorem And Cauchy's Integral Theorem

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

$$\int_{\gamma}f(z)dz=0$$

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?

• What's the question? – user228113 Jun 16 '17 at 9:16
• 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. – Count Iblis Jun 16 '17 at 9:18
• @G.Sassatelli Sorry, added the question – gbox Jun 16 '17 at 9:22

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