In a book (from A. Lesfari, a French author ; "Variables complexes", page 15. Can be found on LibGen), I can read this Lemma :
Let $f$ be a holomorphic function. Thus, we have : $\displaystyle\int_\gamma f'(z)\text{d}z=f(\gamma(b))-f(\gamma(a))$, where $\gamma:[a,b]\to\mathbb{C}$ is $\mathscr{C}^1$.
There is something I cannot understand : consider $f\equiv\log$ on the domain $\mathcal{C}=\lbrace z\in\mathbb{C}\;/\;0.5<|z|<2\rbrace$ and $\gamma:t\in[0,2\pi]\mapsto e^{it}$. We have :
$\displaystyle\int_{S^1}\frac{\text{d}z}{z}=\int_0^{2\pi}\frac1{e^{it}}\times ie^{it}\text{d}z=2\pi i$
But on the other hand, $\log(e^{2\pi i})-\log(e^{0})=0$...
There must be something I'm doing wrong. There is no additionnal requirement on the domain of $f$ (the author does not need it to be simply connected)... The proof of the lemma holds if you notice that $\displaystyle\frac{\text{d}}{\text{d}t}[f(\gamma(t))]=\gamma'(t)f'(\gamma(t))$.
Thanks already for your help :)