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polylog

general polylogarithm function

 Calling Sequence polylog(a, z)

Parameters

 a - expression z - expression

Description

 • The polylogarithm of index a at the point z is defined by

$\mathrm{polylog}\left(a,z\right)=\sum _{n=1}^{\mathrm{\infty }}\frac{{z}^{n}}{{n}^{a}}$

 if $\left|z\right|<1$ and by analytic continuation otherwise.  The index a can be any complex number.  If $\Re \left(a\right)\le 1$, the point $z=1$ is a singularity.
 • For all indices a, the point $z=1$ is a branch point for all branches, and in Maple, the branch cut is taken to be the interval ($1,\mathrm{\infty }$).  For the branches other than the principal branch (which is given on the unit disk by the series above, and hence is analytic at 0), the point $z=0$ is also a branch point, and the branch cut is taken to be the negative real axis.  The formula for a particular branch can be determined with the following rules:
 Each time the branch cut ($1,\mathrm{\infty }$) is crossed in the counterclockwise direction, subtract $\frac{2I\mathrm{\pi }{\mathrm{ln}\left(z\right)}^{a-1}}{\mathrm{\Gamma }\left(a\right)}$. Add this quantity if the branch cut is crossed in the clockwise direction.
 Each time the branch cut ($-\mathrm{\infty },0$) is crossed in the counterclockwise direction, add $2I\mathrm{\pi }$ to each $\mathrm{ln}\left(z\right)$ term in the current formula.  Subtract this quantity if the branch cut is crossed in the clockwise direction.
 For example, if one traverses a path which starts at $z=\frac{1}{2}$, goes clockwise around $z=1$, then counterclockwise around $z=0$, then clockwise around $z=1$ again to return at $z=\frac{1}{2}$, the formula for the branch of polylog thus obtained would be

${\mathrm{Li}}_{a}\left(z\right)+\frac{2I\mathrm{\pi }\left({\mathrm{ln}\left(z\right)}^{a-1}+{\left(\mathrm{ln}\left(z\right)+2I\mathrm{\pi }\right)}^{a-1}\right)}{\mathrm{\Gamma }\left(a\right)}$

 where polylog(a, z) indicates the principal branch and $\mathrm{ln}\left(z\right)$ means the principal branch of the logarithm.
 • Maple only evaluates the principal branch.
 • Maple's dilog function is related to polylog by the relation $\mathrm{dilog}\left(z\right)=\mathrm{polylog}\left(2,1-z\right)$.

Examples

 > $\mathrm{polylog}\left(a,0\right)$
 ${0}$ (1)
 > $\mathrm{polylog}\left(2,1\right)$
 $\frac{{1}}{{6}}{}{{\mathrm{π}}}^{{2}}$ (2)
 > $\mathrm{polylog}\left(3,1\right)$
 ${\mathrm{ζ}}{}\left({3}\right)$ (3)
 > $\mathrm{polylog}\left(2,I\right)$
 ${-}\frac{{1}}{{48}}{}{{\mathrm{π}}}^{{2}}{+}{I}{}{\mathrm{Catalan}}$ (4)
 > $\frac{\partial }{\partial x}\mathrm{polylog}\left(a,x\right)$
 $\frac{{\mathrm{polylog}}{}\left({a}{-}{1}{,}{x}\right)}{{x}}$ (5)
 > $\mathrm{combine}\left(\mathrm{polylog}\left(a,x\right)+\mathrm{polylog}\left(a,-x\right),\mathrm{polylog}\right)$
 ${{2}}^{{1}{-}{a}}{}{\mathrm{polylog}}{}\left({a}{,}{{x}}^{{2}}\right)$ (6)
 > $\mathrm{polylog}\left(4,x\right)+\mathrm{polylog}\left(4,\frac{1}{x}\right)$
 ${\mathrm{polylog}}{}\left({4}{,}{x}\right){+}{\mathrm{polylog}}{}\left({4}{,}\frac{{1}}{{x}}\right)$ (7)
 > $\mathrm{combine}\left(,\mathrm{polylog}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}assuming\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}1
 ${-}\frac{{1}}{{12}}{}{{\mathrm{ln}}{}\left({-}{x}\right)}^{{2}}{}{{\mathrm{π}}}^{{2}}{-}\frac{{7}}{{360}}{}{{\mathrm{π}}}^{{4}}{-}\frac{{1}}{{24}}{}{{\mathrm{ln}}{}\left({-}{x}\right)}^{{4}}$ (8)
 > $\mathrm{combine}\left(,\mathrm{polylog}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}assuming\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}x::\left(\mathrm{RealRange}\left(-1,1\right)\right)$
 ${-}\frac{{1}}{{12}}{}{{\mathrm{ln}}{}\left({-}\frac{{1}}{{x}}\right)}^{{2}}{}{{\mathrm{π}}}^{{2}}{-}\frac{{7}}{{360}}{}{{\mathrm{π}}}^{{4}}{-}\frac{{1}}{{24}}{}{{\mathrm{ln}}{}\left({-}\frac{{1}}{{x}}\right)}^{{4}}$ (9)
 > $\mathrm{polylog}\left(a,{z}^{5}\right)$
 ${\mathrm{polylog}}{}\left({a}{,}{{z}}^{{5}}\right)$ (10)
 > $\mathrm{expand}\left(\right)$
 $\frac{{1}}{{5}}{}{{5}}^{{a}}{}{\mathrm{polylog}}{}\left({a}{,}{\left({-}{1}\right)}^{{2}{/}{5}}{}{z}\right){+}\frac{{1}}{{5}}{}{{5}}^{{a}}{}{\mathrm{polylog}}{}\left({a}{,}{\left({-}{1}\right)}^{{4}{/}{5}}{}{z}\right){+}\frac{{1}}{{5}}{}{{5}}^{{a}}{}{\mathrm{polylog}}{}\left({a}{,}{-}{z}{}{\left({-}{1}\right)}^{{1}{/}{5}}\right){+}\frac{{1}}{{5}}{}{{5}}^{{a}}{}{\mathrm{polylog}}{}\left({a}{,}{-}{z}{}{\left({-}{1}\right)}^{{3}{/}{5}}\right){+}\frac{{1}}{{5}}{}{{5}}^{{a}}{}{\mathrm{polylog}}{}\left({a}{,}{z}\right)$ (11)
 > $x≔'x':$
 > $\mathrm{polylog}\left(1,x\right)$
 ${-}{\mathrm{ln}}{}\left({1}{-}{x}\right)$ (12)
 > $\mathrm{polylog}\left(2,\frac{1}{3}\right)$
 ${\mathrm{polylog}}{}\left({2}{,}\frac{{1}}{{3}}\right)$ (13)
 > $\mathrm{evalf}\left(\right)$
 ${0.3662132299}$ (14)
 > $\mathrm{polylog}\left(-3.7+2.2I,1.5+2.7I\right)$
 ${-}{188.9091729}{+}{104.0046999}{}{I}$ (15)

References

 Lewin, L. Polylogarithms and Associated Functions. Amsterdam: North Holland, 1981.

 See Also