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variants of this functions
PolyLog






Mathematica Notation

Traditional Notation









Zeta Functions and Polylogarithms > PolyLog[2,z] > Transformations > Multiple arguments





http://functions.wolfram.com/10.07.16.0002.01









  


  










Input Form





PolyLog[2, z w] == PolyLog[2, w] + PolyLog[2, z] + PolyLog[2, (z (w - 1))/(1 - z)] + PolyLog[2, (w (z - 1))/(1 - w)] + (1/2) Log[(1 - w)/(1 - z)]^2 /; !(Element[z, Reals] && Element[w, Reals] && z > 1 && w > 1 && (z (w - 1))/(1 - z) > 1 && (w (z - 1))/(1 - w) > 1) || ((Element[z, Reals] && Element[w, Reals] && (z > 1 && w > 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["z", " ", RowBox[List["(", RowBox[List["w", "-", "1"]], ")"]]]], RowBox[List["1", "-", "z"]]]]], "]"]], "+", RowBox[List["PolyLog", "[", RowBox[List["2", ",", FractionBox[RowBox[List["w", " ", RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]]]], RowBox[List["1", "-", "w"]]]]], "]"]], "+", RowBox[List[FractionBox["1", "2"], SuperscriptBox[RowBox[List["Log", "[", FractionBox[RowBox[List["1", "-", "w"]], RowBox[List["1", "-", "z"]]], "]"]], "2"]]]]]]], "/;", RowBox[List[RowBox[List["Not", "[", RowBox[List[RowBox[List["Element", "[", RowBox[List["z", ",", "Reals"]], "]"]], "\[And]", RowBox[List["Element", "[", RowBox[List["w", ",", "Reals"]], "]"]], "\[And]", RowBox[List["z", ">", "1"]], "\[And]", RowBox[List["w", ">", "1"]], "\[And]", RowBox[List[FractionBox[RowBox[List["z", " ", RowBox[List["(", RowBox[List["w", "-", "1"]], ")"]]]], RowBox[List["1", "-", "z"]]], ">", "1"]], "\[And]", RowBox[List[FractionBox[RowBox[List["w", " ", RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]]]], RowBox[List["1", "-", "w"]]], ">", "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["z", ">", "1"]], "\[And]", RowBox[List["w", ">", "1"]]]], ")"]]]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List["z", "<", "1"]], "\[And]", RowBox[List["w", "<", "1"]]]], ")"]]]], ")"]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mn> 2 </mn> </msub> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mi> w </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mn> 2 </mn> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mn> 2 </mn> </msub> <mo> ( </mo> <mi> w </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mn> 2 </mn> </msub> <mo> ( </mo> <mfrac> <mrow> <mi> w </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> w </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mn> 2 </mn> </msub> <mo> ( </mo> <mfrac> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> w </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> w </mi> </mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> &#172; </mo> <mrow> <mo> ( </mo> <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> <mi> z </mi> <mo> &gt; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> w </mi> <mo> &gt; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mfrac> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> w </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> </mfrac> <mo> &gt; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mfrac> <mrow> <mi> w </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> w </mi> </mrow> </mfrac> <mo> &gt; </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </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> <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> &#8744; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <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> ) </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> <ci> PolyLog </ci> <cn type='integer'> 2 </cn> <apply> <times /> <ci> w </ci> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <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> <ci> PolyLog </ci> <cn type='integer'> 2 </cn> <apply> <times /> <ci> z </ci> <apply> <plus /> <ci> w </ci> <cn type='integer'> -1 </cn> </apply> <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='rational'> 1 <sep /> 2 </cn> <apply> <power /> <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> <ci> z </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <or /> <apply> <not /> <apply> <and /> <apply> <in /> <ci> z </ci> <reals /> </apply> <apply> <in /> <ci> w </ci> <reals /> </apply> <apply> <gt /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <gt /> <ci> w </ci> <cn type='integer'> 1 </cn> </apply> <apply> <gt /> <apply> <times /> <ci> z </ci> <apply> <plus /> <ci> w </ci> <cn type='integer'> -1 </cn> </apply> <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> <cn type='integer'> 1 </cn> </apply> <apply> <gt /> <apply> <times /> <ci> w </ci> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <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> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <or /> <apply> <and /> <apply> <in /> <ci> z </ci> <reals /> </apply> <apply> <in /> <ci> w </ci> <reals /> </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> <and /> <apply> <lt /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <lt /> <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["z", " ", RowBox[List["(", RowBox[List["w", "-", "1"]], ")"]]]], RowBox[List["1", "-", "z"]]]]], "]"]], "+", RowBox[List["PolyLog", "[", RowBox[List["2", ",", FractionBox[RowBox[List["w", " ", RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]]]], RowBox[List["1", "-", "w"]]]]], "]"]], "+", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox[RowBox[List["Log", "[", FractionBox[RowBox[List["1", "-", "w"]], RowBox[List["1", "-", "z"]]], "]"]], "2"]]]]], "/;", RowBox[List[RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List["z", "\[Element]", "Reals"]], "&&", RowBox[List["w", "\[Element]", "Reals"]], "&&", RowBox[List["z", ">", "1"]], "&&", RowBox[List["w", ">", "1"]], "&&", RowBox[List[FractionBox[RowBox[List["z", " ", RowBox[List["(", RowBox[List["w", "-", "1"]], ")"]]]], RowBox[List["1", "-", "z"]]], ">", "1"]], "&&", RowBox[List[FractionBox[RowBox[List["w", " ", RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]]]], RowBox[List["1", "-", "w"]]], ">", "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["z", ">", "1"]], "&&", RowBox[List["w", ">", "1"]]]], ")"]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List["z", "<", "1"]], "&&", RowBox[List["w", "<", "1"]]]], ")"]]]], ")"]]]]]]]]]]










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





2001-10-29





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