I'm studying the article by Wood, D.C. "The Computation of Polylogarithms. Technical Report 15-92*" PS (it is remarkably poorly translated from latex to ps). It is listed in the literature section on wiki page on polylogarithm.
In particular, in section $11$ the author studies
$$ -Li_p(-e^w) = \frac{1}{\Gamma(p)}\int_0^{\infty}\frac{t^{p-1}}{e^{t-w}+1}\mathrm dt$$
and then in order to obtain the asymptotics, splits the integral in $w$ (I took the case where $w\in\mathbb R$). Somehow the result is
$$\displaystyle-Li_p(-e^w) = \frac{w^p}{\Gamma(p+1)} + \frac{1}{\Gamma(p)}\int_0^{w}\frac{t^{p-1}}{e^{w-t}+1}\mathrm dt + \frac{1}{\Gamma(p)}\int_w^{\infty}\frac{t^{p-1}}{e^{t-w}+1}\mathrm dt,$$
yet when I do the same calculations, I get
$$-\Gamma(p)Li_p(-e^w) = \int_0^{w}\frac{t^{p-1}}{e^{t-w}+1}\mathrm dt+ \int_w^{\infty}\frac{t^{p-1}}{e^{t-w}+1}\mathrm dt$$ $$= \int_0^{w}\frac{e^{w-t}t^{p-1}}{e^{w-t}+1}\mathrm dt+ \int_w^{\infty}\frac{t^{p-1}}{e^{t-w}+1}\mathrm dt$$ $$= \int_0^{w}\left(1-\frac{1}{e^{w-t}+1}\right)t^{p-1}\mathrm dt+ \int_w^{\infty}\frac{t^{p-1}}{e^{t-w}+1}\mathrm dt\Rightarrow$$ $$ -Li_p(-e^w) = \frac{w^p}{\Gamma(p+1)} \mathbf{-} \frac{1}{\Gamma(p)}\int_0^{w}\frac{t^{p-1}}{e^{w-t}+1}\mathrm dt + \frac{1}{\Gamma(p)}\int_w^{\infty}\frac{t^{p-1}}{e^{t-w}+1}\mathrm dt$$ (note the sign in front of the second term).
Furthermore, their next result reads
$ -Li_p(-e^w) = \frac{w^p}{\Gamma(p+1)} + \frac{1}{\Gamma(p)}\int_0^{\infty}\frac{(w+s)^{p-1}-(w-s)^{p-1}}{e^{s}+1}\mathrm ds - \frac{1}{\Gamma(p)}\int_w^{\infty}\frac{(w-s)^{p-1}}{e^{s}+1}\mathrm ds, $
yet I obtain
$ -Li_p(-e^w) = \frac{w^p}{\Gamma(p+1)} + \frac{1}{\Gamma(p)}\int_0^{\infty}\frac{(w+s)^{p-1}-(w-s)^{p-1}}{e^{s}+1}\mathrm ds + \frac{1}{\Gamma(p)}\int_w^{\infty}\frac{(w-s)^{p-1}}{e^{s}+1}\mathrm ds.$
I start with the previous formula I derived and make two changes of variables $s=w-t$ and $s=t-w$ in second and third terms, respectively.
I see only 3 possible explanations for these disrepancies:
- The translation from LaTeX went so wrong, that the signs $+$ and $-$ went haywire.
- The author of the text made either calculation or typing mistakes
- I've made an error somewhere.
What are your thoughts?