Rearrange Complex Power Series

Question

How do I rearrange this power series: $$\sum_{n=1}^\infty\frac{e^{in}}{n^3}{(z^n-z^{-n})}$$ so that it can be expressed in the form of $$\sum_{n=0}^\infty\ a_n{(z-z_0)}^n $$

Answer 1

At least to me, this is not an easy problem since $$\sum_{n=1}^\infty\frac{e^{in}}{n^3}{(z^n-z^{-n})}=\text{Li}_3\left(e^i z\right)-\text{Li}_3\left(\frac{e^i}{z}\right)$$ Then, the expansion around $z=a$ will include a bunch of polylogarithms everywhere.

The very first terms would be $$\left(\text{Li}_3\left(a e^i\right)-\text{Li}_3\left(\frac{e^i}{a}\right)\right)+\frac{\left(\text{Li}_ 2\left(\frac{e^i}{a}\right)+\text{Li}_2\left(a e^i\right)\right) }{a}(z-a)+$$ $$\frac{ \left(-\text{Li}_2\left(\frac{e^i}{a}\right)-\text{Li}_2\left(a e^i\right)+\log \left(1-\frac{e^i}{a}\right)-\log \left(1-e^i a\right)\right)}{2 a^2}(z-a)^2+O\left((z-a)^3\right)$$