Logs of complex numbers

[lilac stag]

HI i have a quick question, im trying to work out logs of complex numbers. I am familar with how the following formula is derived: $Log(z)= \frac{1}{2}\ln(x^{2}+y^{2})+\left{ i\left( \tan ^{-1}\left( \frac{y}{x})+2k\pi \right) \right) \right}_{k\in \mathbb{Z}}$.

However, i run into an issue if x=0, because then the fraction inside the tan function becomes undefined. What do i do if this is the case?

[dull abyss]

borgg

[raw pine]

I believe $\arctan\left(\frac{y}{x}\right)=\text{sgn}(y)\frac{\pi}{2}$ when $x=0$

[dull abyss]

SWR