Are there any specific notations for complex logarithms in different branch cuts? The principal branch is denoted as $\operatorname{Log}z$. But how do I write a logarithm which is sliced on the negative imaginary axis?
Edit: If there are none, how would you write it? Maybe something like $\log_w^{\arg}$? This looks not good in my opinion that's why I need your advice.
I do not think that there is any widely used notation. However, you can distinguish different branches by specifying the domain of the function. This means that, if the branch you are dealing with has domain $U$, then you can write $\log_U$ or explicitely say that $\log : U \rightarrow \mathbb{C}$. For instance, the principal branch of $\log$ is obtained by using $U=\mathbb{C} \setminus \mathbb{R}_{\leq 0}$ as domain.