A dilogarithm identity (simplification/compaction)

Question

I'm wondering if there is any compact expression to compute (or approximate):

$$\operatorname{Li}_2(pe^{-\alpha})-\operatorname{Li}_2(pe^{\alpha})$$

or

$$\operatorname{Re}\{\operatorname{Li}_2(pe^{-\alpha})-\operatorname{Li}_2(pe^{\alpha})\}$$

The problem is that I cannot use

$$\operatorname{Li}_2(z)=\sum_{n=1}^{\infty}\frac{z^n}{n^2}$$

as for my values, $|z|$ is NOT necessarily less than 1. I am hoping, maybe, having exponential inputs to $\operatorname{Li}_2$ might do some magic. Unfortunately, $p$ and $\alpha$ are pure real. A nice approximation or any piece of advice is greatly appreciated.

Answer 1

For $|z|>1$ you can use the relationship : $$\text{Li}_2(z)= -\frac{1}{2}\ln^2(-z)-\frac{\pi^2}{6}-\text{Li}_2\left(\frac{1}{z} \right)$$ There is no particular difficulty to compute the real value of $-\frac{1}{2}\ln^2(-z)$ .