Playing around with the complex integretion I encountered the following:
Consider a holomorphic function $f(z)$ on $\Omega$. Let's say this holomorphic function has a primitve $F(z)$ such that $F'(z) = f(z)$ on $\Omega$.
Then for each closed $\Gamma\subseteq \Omega$, since $f(z)$ is holomorphic on $\Omega$
$$\begin{align} \int_\Gamma f(z)\text{d} z &= 0\\ \int_\Gamma F'(z) \text{d} z & = 0\\ F(z) &\color{red}{\stackrel{?}{=}} C \end{align}$$
Where $C$ denotes a complex number.
Really?
I don't think this is right, is it? But what's wrong with the argument?