If someone could explain branch cuts and branch points to me that would be fantastic. I understand that a branch cut is a curve that we remove from the domain to make a function (usually a logarithm) analytic. A branch point is a point common to all possible branch cuts.
In the context of the question I'm looking at, I'm not sure I understand how to apply this definition to find a branch cut.
"Find a branch of $\log (iz)$ which is analytic in the region $\{z:\mathrm{Im}(z)>0\}$."
Clearly, it's not the principle logarithm, $\mathrm{Log}$ as this needs a branch cut along the positive imaginary axis to be analytic. I've applied the definition of a complex logarithm as follows:
\begin{align*} \log(iz) =& \log\left(re^{i\left(\theta+\frac{\pi}{2}\right)}\right) \\ =&r + i\left(\theta+\frac{\pi}{2}+2n\pi\right). \end{align*}
Since I know that the origin is a branch point, I must need to get rid of a line passing through the origin of the form $y=mx$. However, from what I have above, I'm not quite sure what to do. There doesn't seem to be any part where there would be a singularity or other problem that would cause the function to be non-analytic. This is how I've been finding other branch cuts - just looking at the $\mathrm{Log}$ function and seeing where the input generates the negative real axis, as I know that this makes the principal log non-analytic. Any help would be appreciated!