# Polylogarithm - derivative with respect to order

Does anybody know where I could find the expression for $$\frac{\partial}{\partial s}\mathrm{Li}_s(z)\bigg|_{s=0}$$ or something similar?

• Please be more clear. – ABC Mar 21 '13 at 14:16
• A good question. There was a question yesterday where the answer was the derivative of $\mathrm{Li}_s(1/2)$ with respect to $s$. – GEdgar Mar 21 '13 at 14:40
• +1. I never found anything about it. However, we can use integral representations. The problem is that we usually evaluate those integrals in terms of the above derivative. So, it's a closed circle as the dog trying to eat its tail. – Felix Marin Aug 13 '14 at 3:27
• Related to: Find the derivative of a polylogarithm function, URL (version: 2015-03-18): math.stackexchange.com/q/1194499 – Erich Sep 17 '16 at 8:41

For every $|z|<1$, we have $$\frac{\partial}{\partial s}\mathrm{Li}_s(z)\Big|_{s=0} = -\sum_{n=1}^\infty \log (n)\frac{z^n}{n^s} \Big|_{s=0} = -\sum_{n=1}^\infty \log(n)\,z^n$$
Use partial integration, then you can relate the differentiation of $\text{Li}(j,x)$ to $\text{Li}(j,x)$ and $\text{Li}(j-1,x)$.