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variants of this functions
PolyLog






Mathematica Notation

Traditional Notation









Zeta Functions and Polylogarithms > PolyLog[nu,z] > Identities > Functional identities > Trilogarithmical cases > Involving nine trilogarithms





http://functions.wolfram.com/10.08.17.0055.01









  


  










Input Form





PolyLog[3, -x^2] + PolyLog[3, -y^2] + PolyLog[3, -z^2] - 2 PolyLog[3, x y] - 2 PolyLog[3, x z] - 2 PolyLog[3, y z] - 2 PolyLog[3, -(x/y)] - 2 PolyLog[3, -(y/z)] - 2 PolyLog[3, -(x/z)] == (Pi^2/3) Log[-(z/y)] + (1/3) Log[-(z/y)]^3 + Log[-(z/y)]^2 Log[-(x/z)] - 2 PolyLog[3, 1] /; x + y + z == x y z










Standard Form





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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> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mo> - </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mo> - </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mi> x </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mi> x </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mi> y </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mi> x </mi> <mi> y </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mi> x </mi> <mi> z </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mi> y </mi> <mi> z </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 3 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> log </mi> <mn> 3 </mn> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mi> z </mi> <mi> y </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mi> x </mi> <mi> z </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mi> z </mi> <mi> y </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 3 </mn> </mfrac> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mi> z </mi> <mi> y </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mn> 3 </mn> </msub> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> x </mi> <mo> + </mo> <mi> y </mi> <mo> + </mo> <mi> z </mi> </mrow> <mo> &#10869; </mo> <mrow> <mi> x </mi> <mo> &#8290; </mo> <mi> y </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <plus /> <apply> <ci> PolyLog </ci> <cn type='integer'> 3 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <ci> PolyLog </ci> <cn type='integer'> 3 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <ci> PolyLog </ci> <cn type='integer'> 3 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> PolyLog </ci> <cn type='integer'> 3 </cn> <apply> <times /> <ci> x </ci> <ci> y </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> PolyLog </ci> <cn type='integer'> 3 </cn> <apply> <times /> <ci> x </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> PolyLog </ci> <cn type='integer'> 3 </cn> <apply> <times /> <ci> y </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> PolyLog </ci> <cn type='integer'> 3 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> x </ci> <apply> <power /> <ci> y </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> PolyLog </ci> <cn type='integer'> 3 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> x </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> PolyLog </ci> <cn type='integer'> 3 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> y </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='rational'> 1 <sep /> 3 </cn> <apply> <power /> <apply> <ln /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> z </ci> <apply> <power /> <ci> y </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <apply> <ln /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> x </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <ln /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> z </ci> <apply> <power /> <ci> y </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 3 </cn> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <ln /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> z </ci> <apply> <power /> <ci> y </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> PolyLog </ci> <cn type='integer'> 3 </cn> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <eq /> <apply> <plus /> <ci> x </ci> <ci> y </ci> <ci> z </ci> </apply> <apply> <times /> <ci> x </ci> <ci> y </ci> <ci> z </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["PolyLog", "[", RowBox[List["3", ",", RowBox[List["-", SuperscriptBox["x_", "2"]]]]], "]"]], "+", RowBox[List["PolyLog", "[", RowBox[List["3", ",", RowBox[List["-", SuperscriptBox["y_", "2"]]]]], "]"]], "+", RowBox[List["PolyLog", "[", RowBox[List["3", ",", RowBox[List["-", SuperscriptBox["z_", "2"]]]]], "]"]], "-", RowBox[List["2", " ", RowBox[List["PolyLog", "[", RowBox[List["3", ",", RowBox[List["x_", " ", "y_"]]]], "]"]]]], "-", RowBox[List["2", " ", RowBox[List["PolyLog", "[", RowBox[List["3", ",", RowBox[List["x_", " ", "z_"]]]], "]"]]]], "-", RowBox[List["2", " ", RowBox[List["PolyLog", "[", RowBox[List["3", ",", RowBox[List["y_", " ", "z_"]]]], "]"]]]], "-", RowBox[List["2", " ", RowBox[List["PolyLog", "[", RowBox[List["3", ",", RowBox[List["-", FractionBox["x_", "y_"]]]]], "]"]]]], "-", RowBox[List["2", " ", RowBox[List["PolyLog", "[", RowBox[List["3", ",", RowBox[List["-", FractionBox["y_", "z_"]]]]], "]"]]]], "-", RowBox[List["2", " ", RowBox[List["PolyLog", "[", RowBox[List["3", ",", RowBox[List["-", FractionBox["x_", "z_"]]]]], "]"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[FractionBox["1", "3"], " ", SuperscriptBox["\[Pi]", "2"], " ", RowBox[List["Log", "[", RowBox[List["-", FractionBox["z", "y"]]], "]"]]]], "+", RowBox[List[FractionBox["1", "3"], " ", SuperscriptBox[RowBox[List["Log", "[", RowBox[List["-", FractionBox["z", "y"]]], "]"]], "3"]]], "+", RowBox[List[SuperscriptBox[RowBox[List["Log", "[", RowBox[List["-", FractionBox["z", "y"]]], "]"]], "2"], " ", RowBox[List["Log", "[", RowBox[List["-", FractionBox["x", "z"]]], "]"]]]], "-", RowBox[List["2", " ", RowBox[List["PolyLog", "[", RowBox[List["3", ",", "1"]], "]"]]]]]], "/;", RowBox[List[RowBox[List["x", "+", "y", "+", "z"]], "\[Equal]", RowBox[List["x", " ", "y", " ", "z"]]]]]]]]]]










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





2001-10-29





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