# Question regarding assumptions in Morera's theorem and conditions for existence of antiderivative.

Morera's Theorem as it is phrased in Wikipedia states that if $\Omega\subseteq\mathbb{C}$ is a simply connected open set in the plain and $f:\Omega\to\mathbb{C}$ is a continuous function such that for every closed piece-wise $C^{1}$ curve in $\Omega$ the following equality holds: $$\left(\star\right)\;\oint_{\gamma}f\left(z\right)dz=0$$ Then $f$ is holomorphic in $\Omega$. The proof in this case follows by constructing an anti-derivative of $f$ in $\Omega$ and deducing that $f$ is holomorphic as the derivative of a holomorphic function. My question is this: Suppose we omit the assumption that $\Omega$ is simply connected (but remains connected) and assume $f$ is a function such that for every aforementioned $\gamma$ the equality $\left(\star\right)$ holds. Pick any $z_{0}\in\Omega$ and define: $$F\left(z\right)=\int\limits _{z_{0}}^{z}f\left(\omega\right)d\omega$$ Wouldn't it follow that $F$ is an antiderivative of $f$ in $\Omega$ ? I believe the definition would be independent of the choice of path of integration from $z_{0}$ to $z$ from the assumption $\left(\star\right)$ holds.

• @AmritanshuPrasad I don't see why F would be not well-defined. An open connected set is also path connected, so there is no problem to define F. Commented Jan 16, 2014 at 9:27
• @Serpahimz if I remember well my complex analysis: If $\Omega$ is an open connected set then "Every closed contour = 0" is equivalent to say that "Contours depens only on endpoints" thus proving that your right Commented Jan 16, 2014 at 9:31
• Oh sorry, you are right. I was not reading this carefully. Of course, that $\Omega$ is simply connected is needed in the proof only to establish that the integral of $f$ does not depend on the path. If we already know that, we do not need this assumption. In fact, such an $f$ can be extended to a simply connected domain containing $\Omega$. Commented Jan 16, 2014 at 9:36
• You are correct. Wikipedia is deficient on this topic. mathworld.wolfram.com/MorerasTheorem.html Commented Jan 16, 2014 at 9:37