What calculate $\ln i$ I would like to know how to calculate $\ln i$. I found a formula on the internet that says $$\ln z=\ln|z|+i\operatorname{Arg}(z)$$Then $|i|=1$ and $\operatorname{Arg}(i)$ is?
 A: If you want to know what is $\ln i$ you should make yourself aware of what the result should be. By definition the logarithm of $i$ should be some complex number $z$ such that $e^z=i$. But by Euler's formula $e^{i\pi/2}=\cos(\pi/2)+i\sin(\pi/2)=i$, so you could say that "$\ln i =i\pi/2$". And that is true if we choose $\ln$ to be the principal branch of the complex logarithm.
But be aware that since the exponential function is periodic, also $e^{i\pi/2+2\pi ik}=i$ holds for all $k\in\mathbb{Z}$. Therefore the logarithm is a multi-valued function.
A: Note that you are referencing the argument with capital $A$, which implies the principal value, i.e., $\operatorname{Arg} z \in (-\pi,\pi]$.  Therefore $\operatorname{Arg}(i) = \pi/2$ and $\log i = i \pi/2$.
A: Given any non-zero complex number $z,$ $\operatorname{Arg}(z)$ (the principal argument of $z$) is the unique $\theta\in(-\pi,\pi]$ such that $z=|z|e^{i\theta}.$ Observing that $\theta$ gives a radian measure of rotation from the positive real axis, what must the principal argument of $i$ be, that is, how much must we rotate $|i|=1$ to get it to $i$ again?
