combine/polylog - Maple Help

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combine/polylog

combine polylog functions

 Calling Sequence combine(f, polylog);

Parameters

 f - any expression

Description

 Expressions involving polylog are combined as follows

$\mathrm{polylog}\left(n,x\right)+{\left(-1\right)}^{n}\mathrm{polylog}\left(n,\frac{1}{x}\right)=2\left(\sum _{k=1}^{\mathrm{iquo}\left(n,2\right)}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{\mathrm{ln}\left(-x\right)}^{n-2k}\mathrm{polylog}\left(2k,-1\right)}{\left(n-2k\right)!}\right)$

$\mathrm{polylog}\left(n,\frac{1}{y}\right)+{\left(-1\right)}^{n}\mathrm{polylog}\left(n,y\right)=2{\left(-1\right)}^{n}\left(\sum _{k=1}^{\mathrm{iquo}\left(n,2\right)}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{\mathrm{ln}\left(-\frac{1}{y}\right)}^{n-2k}\mathrm{polylog}\left(2k,-1\right)}{\left(n-2k\right)!}\right)$

$\mathrm{polylog}\left(a,z\right)+\mathrm{polylog}\left(a,-z\right)={2}^{1-a}\mathrm{polylog}\left(a,{z}^{2}\right)$

$\mathrm{polylog}\left(2,z\right)+\mathrm{polylog}\left(2,1-z\right)=\frac{{\mathrm{\pi }}^{2}}{6}-\mathrm{ln}\left(z\right)\mathrm{ln}\left(1-z\right)$

$\mathrm{polylog}\left(2,w\right)+\mathrm{polylog}\left(2,\frac{w}{w-1}\right)=\left\{\begin{array}{cc}-\frac{1}{2}{\mathrm{ln}\left(1-w\right)}^{2}& w\le 1\\ \frac{1}{2}{\mathrm{\pi }}^{2}-2I\mathrm{\pi }\mathrm{ln}\left(w\right)+I\mathrm{\pi }\mathrm{ln}\left(w-1\right)-\frac{1}{2}{\mathrm{ln}\left(w-1\right)}^{2}& 1

$\mathrm{polylog}\left(3,w\right)+\mathrm{polylog}\left(3,1-w\right)+\mathrm{polylog}\left(3,\frac{w}{w-1}\right)=\mathrm{polylog}\left(3,1\right)+\frac{1}{6}{\mathrm{\pi }}^{2}\mathrm{ln}\left(1-w\right)-\frac{1}{2}\mathrm{ln}\left(w\right){\mathrm{ln}\left(1-w\right)}^{2}+\frac{1}{6}{\mathrm{ln}\left(1-w\right)}^{3},\mathrm{for}w<1$

 where n is an integer greater than one, a, w, x, y, and z are variables assumed to be complex, where $1\le \left|x\right|$, and $\left|y\right|<1$.

Examples

 > $\mathrm{combine}\left(\mathrm{polylog}\left(a,x\right)+\mathrm{polylog}\left(a,-x\right),\mathrm{polylog}\right)$
 ${{2}}^{{1}{-}{a}}{}{\mathrm{polylog}}{}\left({a}{,}{{x}}^{{2}}\right)$ (1)
 > $\mathrm{assume}\left(1
 > $\mathrm{combine}\left(\mathrm{polylog}\left(2,x\right)+\mathrm{polylog}\left(2,\frac{x}{x-1}\right),\mathrm{polylog}\right)$
 $\frac{{1}}{{2}}{}{{\mathrm{π}}}^{{2}}{-}{2}{}{I}{}{\mathrm{π}}{}{\mathrm{ln}}{}\left({\mathrm{x~}}\right){+}{I}{}{\mathrm{π}}{}{\mathrm{ln}}{}\left({\mathrm{x~}}{-}{1}\right){-}\frac{{1}}{{2}}{}{{\mathrm{ln}}{}\left({\mathrm{x~}}{-}{1}\right)}^{{2}}$ (2)
 > $\mathrm{assume}\left(x,\mathrm{RealRange}\left(-1,1\right)\right)$
 > $\mathrm{combine}\left(\mathrm{polylog}\left(4,x\right)+\mathrm{polylog}\left(4,\frac{1}{x}\right),\mathrm{polylog}\right)$
 ${-}\frac{{1}}{{12}}{}{{\mathrm{ln}}{}\left({-}\frac{{1}}{{\mathrm{x~}}}\right)}^{{2}}{}{{\mathrm{π}}}^{{2}}{-}\frac{{7}}{{360}}{}{{\mathrm{π}}}^{{4}}{-}\frac{{1}}{{24}}{}{{\mathrm{ln}}{}\left({-}\frac{{1}}{{\mathrm{x~}}}\right)}^{{4}}$ (3)
 > $\mathrm{assume}\left(x<1\right)$
 > $\mathrm{combine}\left(\mathrm{polylog}\left(3,x\right)+\mathrm{polylog}\left(3,1-x\right)+\mathrm{polylog}\left(3,\frac{x}{x-1}\right),\mathrm{polylog}\right)$
 ${\mathrm{ζ}}{}\left({3}\right){+}\frac{{1}}{{6}}{}{{\mathrm{π}}}^{{2}}{}{\mathrm{ln}}{}\left({1}{-}{\mathrm{x~}}\right){-}\frac{{1}}{{2}}{}{\mathrm{ln}}{}\left({\mathrm{x~}}\right){}{{\mathrm{ln}}{}\left({1}{-}{\mathrm{x~}}\right)}^{{2}}{+}\frac{{1}}{{6}}{}{{\mathrm{ln}}{}\left({1}{-}{\mathrm{x~}}\right)}^{{3}}$ (4)

 See Also

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