I'm struggling to understand the theory behind branch cuts; what are they, why do we do them, etc. The following problem is directly from my homework.
Find the derivative of $f(z) = Log(z)$, recall that the constraint must be written, and determine $Log[f'(−i)]$.
So, I guess, my first question is regarding log vs. Log. I understand the point of the principal value, but how does it apply to the log function?
Secondly, the solution to this problem states that the constraint is $-i = e^{-i \frac{\pi}{2}}$. I understand how to get from $-i$ to $e^{-i \frac{\pi}{2}}$, but I don't understand why this branch cut was chosen.
I can perform the derivative and solve for $f'(-i)$ parts easy enough. I'm just unsure of the work involving the constraints (which I'm assuming is the branch cut).
Thank you.