I need to introduce a pair of invertible complex functions $\exp$ and $\log$ with the following properties:
$A$ being a branch (or strip?) of $\mathbb{C} \backslash \{0\}$:
$\forall a \in A \quad \exp(\log(a)) = \log(\exp(a)) = a$
$\forall a \in \mathbb{R}^* \quad \log(-|a|) = \log(|a|) + i\pi$
What is the proper way of introducing these two functions? I am especially concerned about the proper enunciation of the functions’ domains and codomains.
Furthermore, is there a proper way of introducing the “continuation” of the first property to a subset containing 0?
I am somewhat familiar with complex logarithms, but I still struggle with the proper definition of domains and codomains using branches (strips?). I would like the definitions to be as precise and unambiguous as can be.