As stated in the question, the set $A$ is $z: 0<|z|<3$ and $f(z)=\frac{z}{z+3}$ I know it does not, it is easy to show by taking derivative of $Logf(z)$ and calculating the integral on circle (0,1), which is not zero from cauchy theorem-> so the derivative has no primary function etc. But my question is: 1) Is set $A$ simply connected space? If so, I have read that the logarithm has contionous branch on the simply connected space, when f(z) is not equal to zero. Is it true? There seems to be contradiction here.
Thanks for any explanation.
$A$ is not simply connected, and $\log f$ cannot be defined on it in the way you ask because of the singularity at $z=0$. Note that had the question taken $\mathbb C$ and removed the slit $[-3,0]$, then a continuous branch would be definable, because as you circle $0$ and $-3$ the extra $2\pi i$'s that you clock up from each log function cancel. Indeed, the function $\log \frac{z}{z+3}$ is regular at $\infty$.