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variants of this functions
PolyLog




Mathematica Notation

Traditional Notation









Zeta Functions and Polylogarithms > PolyLog[2,z] > Identities > Functional identities > Involving five dilogarithms





http://functions.wolfram.com/10.07.17.0019.01









  


  








Input Form





PolyLog[2, (z/(1 - z)) (w/(1 - w))] == PolyLog[2, z/(1 - w)] + PolyLog[2, w/(1 - z)] - PolyLog[2, z] - PolyLog[2, w] + (1/2) ((-Log[1 - w]) Log[w] - Log[1 - z] Log[z] + Log[z/(1 - w)] Log[(-1 + w + z)/(-1 + w)] + Log[w/(1 - z)] Log[(-1 + w + z)/(-1 + z)] - Log[(w z)/((-1 + w) (-1 + z))] Log[-((-1 + w + z)/((-1 + w) (-1 + z)))]) /; (Abs[z] < 1 && 0 < w < 1) || (Abs[w] < 1 && 0 < z < 1) || (z < 1 && 0 < w < 1) || (w < 1 && 0 < z < 1) || (z + w > 1 && w < 0) || (z + w > 1 && z < 0)








Standard Form





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MathML Form







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</mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> z </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &lt; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mn> 0 </mn> <mo> &lt; </mo> <mi> w </mi> <mo> &lt; </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8744; </mo> <mrow> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> w </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &lt; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mn> 0 </mn> <mo> &lt; </mo> <mi> z </mi> <mo> &lt; </mo> <mn> 1 </mn> </mrow> </mrow> <mo> &#8744; </mo> <mrow> <mrow> <mi> z </mi> <mo> &lt; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mn> 0 </mn> <mo> &lt; </mo> <mi> w </mi> <mo> &lt; </mo> <mn> 1 </mn> </mrow> </mrow> <mo> &#8744; </mo> <mrow> <mrow> <mi> w </mi> <mo> &lt; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mn> 0 </mn> <mo> &lt; </mo> <mi> z </mi> <mo> &lt; </mo> <mn> 1 </mn> </mrow> </mrow> <mo> &#8744; </mo> <mrow> <mrow> <mrow> <mi> w </mi> <mo> + </mo> <mi> z </mi> </mrow> <mo> &gt; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> w </mi> <mo> &lt; </mo> <mn> 0 </mn> </mrow> </mrow> <mo> &#8744; </mo> <mrow> <mrow> <mrow> <mi> w </mi> <mo> + </mo> <mi> z </mi> </mrow> <mo> &gt; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> z </mi> <mo> &lt; </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> PolyLog </ci> <cn type='integer'> 2 </cn> <apply> <times /> <ci> z </ci> <ci> w </ci> <apply> <power /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <plus /> <apply> <ci> PolyLog </ci> <cn type='integer'> 2 </cn> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <ci> PolyLog </ci> <cn type='integer'> 2 </cn> <apply> <times /> <ci> w </ci> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyLog </ci> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyLog </ci> <cn type='integer'> 2 </cn> <ci> w </ci> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <apply> <ln /> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ln /> <apply> <times /> <apply> <plus /> <ci> w </ci> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> w </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <ln /> <apply> <times /> <ci> w </ci> <apply> <power /> <apply> <plus /> 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/> <ci> w </ci> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> <apply> <ln /> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> </apply> <apply> <ln /> <ci> w </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <or /> <apply> <and /> <apply> <lt /> <apply> <abs /> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <lt /> <cn type='integer'> 0 </cn> <ci> w </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <and /> <apply> <lt /> <apply> <abs /> <ci> w </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <lt /> <cn type='integer'> 0 </cn> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <and /> <apply> <lt /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <lt /> <cn type='integer'> 0 </cn> <ci> w </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <and /> <apply> <lt /> <ci> w </ci> <cn type='integer'> 1 </cn> </apply> <apply> <lt /> <cn type='integer'> 0 </cn> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <and /> <apply> <gt /> <apply> <plus /> <ci> w </ci> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <lt /> <ci> w </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <and /> <apply> <gt /> <apply> <plus /> <ci> w </ci> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <lt /> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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Date Added to functions.wolfram.com (modification date)





2001-10-29