Function: polylog
Section: transcendental
C-Name: polylog0
Prototype: LGD0,L,p
Help: polylog(m,x,{flag=0}): m-th polylogarithm of x. flag is optional, and
 can be 0: default, 1: D_m~-modified m-th polylog of x, 2: D_m-modified m-th
 polylog of x, 3: P_m-modified m-th polylog of x.
Doc: one of the different polylogarithms, depending on $\fl$:

 If $\fl=0$ or is omitted: $m^{\text{th}}$ polylogarithm of $x$, i.e.~analytic
 continuation of the power series $\text{Li}_{m}(x)=\sum_{n\ge1}x^{n}/n^{m}$
 ($x < 1$). Uses the functional equation linking the values at $x$ and $1/x$
 to restrict to the case $|x|\leq 1$, then the power series when
 $|x|^{2}\le1/2$, and the power series expansion in $\log(x)$ otherwise.

 Using $\fl$, computes a modified $m^{\text{th}}$ polylogarithm of $x$.
 We use Zagier's notations; let $\Re_{m}$ denote $\Re$ or $\Im$ depending
 on whether $m$ is odd or even:

 If $\fl=1$: compute $\tilde D_{m}(x)$, defined for $|x|\le1$ by
 $$\Re_{m}\left(\sum_{k=0}^{m-1} \dfrac{(-\log|x|)^{k}}{k!}\text{Li}_{m-k}(x)
 +\dfrac{(-\log|x|)^{m-1}}{m!}\log|1-x|\right).$$

 If $\fl=2$: compute $D_{m}(x)$, defined for $|x|\le1$ by
 $$\Re_{m}\left(\sum_{k=0}^{m-1}\dfrac{(-\log|x|)^{k}}{k!}\text{Li}_{m-k}(x)
 -\dfrac{1}{2}\dfrac{(-\log|x|)^{m}}{m!}\right).$$

 If $\fl=3$: compute $P_{m}(x)$, defined for $|x|\le1$ by
 $$\Re_{m}\left(\sum_{k=0}^{m-1}\dfrac{2^{k}B_{k}}{k!}
   (\log|x|)^{k}\text{Li}_{m-k}(x)
 -\dfrac{2^{m-1}B_{m}}{m!}(\log|x|)^{m}\right).$$

 These three functions satisfy the functional equation
   $f_{m}(1/x) = (-1)^{m-1}f_{m}(x)$.
Variant: Also available is
 \fun{GEN}{gpolylog}{long m, GEN x, long prec} ($\fl = 0$).
