I'm reading a PhD dissertation. The author mentioned a branch cut when calculating the multivalued function , where k is a complex number, and he introduced a small number to form . The branch cut is chosen as shown in the figure attached. And the author then asserts that the branch cut so chosen is because he wants to have when approaches zero, when and when .
My question is how does the author get this conclusion? i.e., when and when with this branch cut. For another example, the branch cut of would be (-1,1) because in this interval the values are different across the branch cut, while in other intervals like (-infty,-1) and (1, infty) the function is single valued.