This is an issue I am having in Mathematica version 13.1.0.0. According to https://functions.wolfram.com/ElementaryFunctions/Power/04/05/02/, the power function should have a branch cut for $x<0$ described by the following formula:
$$\lim_{a\rightarrow +0}(x-\mathbf{i}\,a)^r = e^{-\mathbf{i} \pi r}(-x)^r$$
The issue I am having is that this formula is not reproduced by the Series[] function. Here is the code I am using:
lim1 = Normal@Series[(x - I a)^r, {a, 0, 0}, Assumptions -> x < 0 && a > 0] $x^r$
It appears mathematica ignores this formula, and effectively just sets $a=0$. This is not an issue with the Limit[] function, as can be seen with this code:
lim2 = Limit[(x - I a)^r, a -> 0, Direction -> "FromAbove", Assumptions -> x < 0] $\mathbf{e}^{-\mathbf{i}\pi r}(-x)^r$
I also compared the two answers numerically for $x=-1$ and $r=1/2$, essentially examining the branch cut of the square root function. Here is that code:
{lim1, lim2, (x - I a)^r} /. {x -> -1, r -> 1/2, a -> 2.^-100} $\left\{ \mathbf{i},\; -\mathbf{i},\; 6.12323\times10^{-17} - 1.\;\mathbf{i} \right\}$
So again, using Limit[] seems to agree with the numerical approximation, while Series[] does not. Frustratingly, this is in contrast to another problem posted here. In my other problem, Series[] reproduces the branch cut for PolyLog[] correctly, while Limit[] does not. Because of this, I cannot think of a systematic way to deal with the branch cut for expressions like x^r PolyLog[n,-x], which is what I really need.
