This is how I like to think of branch cuts. Consider the function . This function is defined for all as where is the regular natural logarithm and is the argument on . Now, this function is continous everywhereexcepton the line segment . It can be shown that is analytic everywhere except on which we call "its branch". Note, if we defined the argument function on the interval then our newly defined logarithm will have branch but the former is the one usually used. Thus, when the problem asks to find the branch of think of it as asking "where is the function non-analytic". From above, that happens when is on the line segment and that happens when .