I want to determine the Laurent series of the function $$f(z) = \left(\frac{z}{z + x_0}\right)^{\kappa}$$, where $x_0 \in \mathbb R$ and $x_0 > 0$,$\kappa$ is either an integer, a rational number or an irrational number, expanded at $z_0=0$. With some "intuition", I think $z_0 = 0$ is a good point for expansion precisely because at that point $f(z)$ is not defined.
I know that the Laurent series is given by $f(z) = \sum_{n=-\infty}^{\infty}a_n(z-z_0)^n$ which can be split up into the analytic part $\sum_{n=0}^{\infty} a_n (z-z_0)^n$ and the principal part $\sum_{n=1}^{\infty}a_{-n} (z-z_0)^{-n}$.
Because we have a pole at $z= -x_0$, we look at two cases:
- $|z| < x_0$
- $|z| > x_0$
What I managed to do is to determine the Laurent series for $f(z) = \frac{z}{z+x_0}$ and I got $f(z) = \sum_{n=0}^{\infty}(-1)^n \frac{z^{n+1}}{x_0^{n+1}}$ for the first case and $f(z) = \sum_{n=0}^{\infty}(-1)^n \frac{x_0^n}{z^n}$ for the second case. From my understanding, $f(z) = \sum_{n=0}^{\infty}(-1)^n \frac{z^{n+1}}{x_0^{n+1}}$ is the analytic part and $f(z) = \sum_{n=0}^{\infty}(-1)^n \frac{x_0^n}{z^n}$ is the principal part.
Can this result be somehow used for $f(z) = \left(\frac{z}{z + x_0}\right)^{\kappa}$ ?
Another idea was to rewrite the function as $f(z) = e^{\kappa \log(\frac{z}{z+x_0})}$ but I'm not sure how to proceed further.