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variants of this functions
PolyLog




Mathematica Notation

Traditional Notation









Zeta Functions and Polylogarithms > PolyLog[nu,z] > Transformations > Multiple arguments > Dilogarithmical case





http://functions.wolfram.com/10.08.16.0003.01









  


  








Input Form





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








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["z", " ", "w"]]]], "]"]], "\[Equal]", RowBox[List[RowBox[List["PolyLog", "[", RowBox[List["2", ",", "w"]], "]"]], "+", RowBox[List["PolyLog", "[", RowBox[List["2", ",", "z"]], "]"]], "-", RowBox[List["PolyLog", "[", RowBox[List["2", ",", FractionBox[RowBox[List["w", " ", RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]]]], RowBox[List["1", "-", RowBox[List["w", " ", "z"]]]]]]], "]"]], "-", RowBox[List["PolyLog", "[", RowBox[List["2", ",", FractionBox[RowBox[List["z", RowBox[List["(", RowBox[List["1", "-", "w"]], ")"]]]], RowBox[List["1", "-", RowBox[List["w", " ", "z"]]]]]]], "]"]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Log", "[", RowBox[List["1", "-", "w"]], "]"]], " ", RowBox[List["Log", "[", "w", "]"]]]], "+", RowBox[List[RowBox[List["Log", "[", RowBox[List["1", "-", "z"]], "]"]], " ", RowBox[List["Log", "[", "z", "]"]]]], "-", RowBox[List[RowBox[List["Log", "[", RowBox[List["w", " ", "z"]], "]"]], " ", RowBox[List["Log", "[", RowBox[List["1", "-", RowBox[List["w", " ", "z"]]]], "]"]]]], "-", RowBox[List[RowBox[List["Log", "[", FractionBox[RowBox[List["1", "-", "w"]], RowBox[List["1", "-", RowBox[List["w", " ", "z"]]]]], "]"]], " ", RowBox[List["Log", "[", FractionBox[RowBox[List["w", " ", RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]]]], RowBox[List["1", "-", RowBox[List["w", " ", "z"]]]]], "]"]]]], "-", RowBox[List[RowBox[List["Log", "[", FractionBox[RowBox[List["1", "-", "z"]], RowBox[List["1", "-", RowBox[List["w", " ", "z"]]]]], "]"]], " ", RowBox[List["Log", "[", FractionBox[RowBox[List["z", RowBox[List["(", RowBox[List["1", "-", "w"]], ")"]]]], RowBox[List["1", "-", RowBox[List["w", " ", "z"]]]]], "]"]]]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "<", "1"]], "\[And]", RowBox[List["0", "<", "w", "<", "1"]]]], ")"]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "w", "]"]], "<", "1"]], "\[And]", RowBox[List["0", "<", "z", "<", "1"]]]], ")"]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["z", "\[Element]", "Reals"]], "\[And]", RowBox[List["w", "\[Element]", "Reals"]], "\[And]", RowBox[List["(", RowBox[List[RowBox[List["0", "<", "z", "<", "1"]], "\[And]", RowBox[List["w", "<", "1"]]]], ")"]]]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List["0", "<", "w", "<", "1"]], "\[And]", RowBox[List["z", "<", "1"]]]], ")"]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List["z", ">", "1"]], "\[And]", RowBox[List["w", ">", "1"]]]], ")"]]]], ")"]]]]]]]]








MathML Form







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) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> w </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> w </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> w </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> w </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> w </mi> </mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> w </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> w </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> w </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> w </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> w </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <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> <mo> ( </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> ) </mo> </mrow> <mo> &#8744; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> z </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[Reals]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> w </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[Reals]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 0 </mn> <mo> &lt; </mo> <mi> z </mi> <mo> &lt; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> w </mi> <mo> &lt; </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8744; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 0 </mn> <mo> &lt; </mo> <mi> w </mi> <mo> &lt; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> z </mi> <mo> &lt; </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8744; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> &gt; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> w </mi> <mo> &gt; </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </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> </apply> <apply> <plus /> <apply> <ci> PolyLog </ci> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <apply> <ci> PolyLog </ci> <cn type='integer'> 2 </cn> <ci> w </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyLog </ci> <cn type='integer'> 2 </cn> <apply> <times /> <ci> z </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> w </ci> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyLog </ci> <cn type='integer'> 2 </cn> <apply> <times /> <ci> w </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> w </ci> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <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> <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> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ln /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> w </ci> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ln /> <apply> <times /> <ci> z </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> w </ci> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ln /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> w </ci> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ln /> <apply> <times /> <ci> w </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> w </ci> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ln /> <apply> <times /> <ci> w </ci> <ci> z </ci> </apply> </apply> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> w </ci> <ci> z </ci> </apply> </apply> </apply> </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> <or /> <apply> <and /> <apply> <in /> <ci> z </ci> <reals /> </apply> <apply> <in /> <ci> w </ci> <reals /> </apply> <apply> <and /> <apply> <lt /> <cn type='integer'> 0 </cn> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <lt /> <ci> w </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <and /> <apply> <lt /> <cn type='integer'> 0 </cn> <ci> w </ci> <cn type='integer'> 1 </cn> </apply> <apply> <lt /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <and /> <apply> <gt /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <gt /> <ci> w </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["z_", " ", "w_"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["PolyLog", "[", RowBox[List["2", ",", "w"]], "]"]], "+", RowBox[List["PolyLog", "[", RowBox[List["2", ",", "z"]], "]"]], "-", RowBox[List["PolyLog", "[", RowBox[List["2", ",", FractionBox[RowBox[List["w", " ", RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]]]], RowBox[List["1", "-", RowBox[List["w", " ", "z"]]]]]]], "]"]], "-", RowBox[List["PolyLog", "[", RowBox[List["2", ",", FractionBox[RowBox[List["z", " ", RowBox[List["(", RowBox[List["1", "-", "w"]], ")"]]]], RowBox[List["1", "-", RowBox[List["w", " ", "z"]]]]]]], "]"]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Log", "[", RowBox[List["1", "-", "w"]], "]"]], " ", RowBox[List["Log", "[", "w", "]"]]]], "+", RowBox[List[RowBox[List["Log", "[", RowBox[List["1", "-", "z"]], "]"]], " ", RowBox[List["Log", "[", "z", "]"]]]], "-", RowBox[List[RowBox[List["Log", "[", RowBox[List["w", " ", "z"]], "]"]], " ", RowBox[List["Log", "[", RowBox[List["1", "-", RowBox[List["w", " ", "z"]]]], "]"]]]], "-", RowBox[List[RowBox[List["Log", "[", FractionBox[RowBox[List["1", "-", "w"]], RowBox[List["1", "-", RowBox[List["w", " ", "z"]]]]], "]"]], " ", RowBox[List["Log", "[", FractionBox[RowBox[List["w", " ", RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]]]], RowBox[List["1", "-", RowBox[List["w", " ", "z"]]]]], "]"]]]], "-", RowBox[List[RowBox[List["Log", "[", FractionBox[RowBox[List["1", "-", "z"]], RowBox[List["1", "-", RowBox[List["w", " ", "z"]]]]], "]"]], " ", RowBox[List["Log", "[", FractionBox[RowBox[List["z", " ", RowBox[List["(", RowBox[List["1", "-", "w"]], ")"]]]], RowBox[List["1", "-", RowBox[List["w", " ", "z"]]]]], "]"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "<", "1"]], "&&", RowBox[List["0", "<", "w", "<", "1"]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "w", "]"]], "<", "1"]], "&&", RowBox[List["0", "<", "z", "<", "1"]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["z", "\[Element]", "Reals"]], "&&", RowBox[List["w", "\[Element]", "Reals"]], "&&", RowBox[List["(", RowBox[List[RowBox[List["0", "<", "z", "<", "1"]], "&&", RowBox[List["w", "<", "1"]]]], ")"]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List["0", "<", "w", "<", "1"]], "&&", RowBox[List["z", "<", "1"]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List["z", ">", "1"]], "&&", RowBox[List["w", ">", "1"]]]], ")"]]]], ")"]]]]]]]]]]


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





2001-10-29