I want to find the domain of analyticity of the function $f(z)$ = $log$ ($z$ -7 +i) and the derivative over the branch ($\pi$/2,5$\pi$ /2)
I have trouble doing this for the branch mentioned but I know that for the principal logarithm branch $Log(z)$ is analytical in (-$\pi$ ,$\pi$) so the domain of analyticity of the function $Log(g(z))$ where $g(z)$ is analytical will be the points where $g(z)$ is defined and $g(z)$ does not belong to the set { $z =x +iy$ / -$ifnty$ < x <=o and $y=0$ }.
Thus $log$ ($z$ -7 +i) will be analytical over the domain $\mathbb{C}$- { $z =x +iy$/ x<= 7 and y= -1 } Is this thought right? Therefore im wondering what to do if the branch is ($\pi$/2,5$\pi$ /2) or any other non usual branch.
You are right about the domain of analyticity (which you obtain by traslating that of $Log$). Of course other choices can be made, for example the one in the text. The fact is that the value of the function at a certain point is defined only up to an integer multiple of some constant, but the derivative is well-defined and is always $\frac{1}{z}$ (in your case, $\frac{1}{z-7+i}$).