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.