I have a short question. I have the following function $f\left(z\right) = \log\left(z + i\right)$.
The question is, construct a branch cut such that $f\left(z\right)$ is analytic at $f\left(0\right) = i\frac{5}{2}\pi$.
So this is my answer:
We can construct a lot of branch cut. An obvious choice is to remove everything under -i from the domain. So the domain for the function is restricted, the domain of the function is:
$D_{f} = \begin{Bmatrix} z: & \frac{3}{2}\pi < arg \left(z + i\right) < \frac{7}{2}\pi , & |z+i| > 0 \\ \end{Bmatrix}$.
If I am correct, I have now a single valued function which is analytic at $f\left(0\right) = i\frac{5}{2}\pi$.
My question is, is this correct?