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http://functions.wolfram.com/10.08.03.0032.01
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PolyLog[3, E^(-((2 Pi I)/3))] == (-(4/9)) Zeta[3] - (2 I Pi^3)/81
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Cell[BoxData[RowBox[List[RowBox[List["PolyLog", "[", RowBox[List["3", ",", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]"]], "3"]]]]]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["4", "9"]]], " ", RowBox[List["Zeta", "[", "3", "]"]]]], "-", FractionBox[RowBox[List["2", " ", "\[ImaginaryI]", " ", SuperscriptBox["\[Pi]", "3"]]], "81"]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mn> 3 </mn> </msub> <mo> ( </mo> <msup> <mi> ⅇ </mi> <mrow> <mo> - </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> <mn> 3 </mn> </mfrac> </mrow> </msup> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 4 </mn> <mn> 9 </mn> </mfrac> </mrow> <mo> ⁢ </mo> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <semantics> <mn> 3 </mn> <annotation encoding='Mathematica'> TagBox["3", Rule[Editable, True]] </annotation> </semantics> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <msup> <mi> π </mi> <mn> 3 </mn> </msup> </mrow> <mn> 81 </mn> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> PolyLog </ci> <cn type='integer'> 3 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <imaginaryi /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 4 <sep /> 9 </cn> </apply> <apply> <ci> Zeta </ci> <apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <apply> <power /> <pi /> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <cn type='integer'> 81 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["PolyLog", "[", RowBox[List["3", ",", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "3"]]], " ", RowBox[List["(", RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]"]], ")"]]]]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "9"], " ", RowBox[List["(", RowBox[List["-", "4"]], ")"]], " ", RowBox[List["Zeta", "[", "3", "]"]]]], "-", FractionBox[RowBox[List["2", " ", "\[ImaginaryI]", " ", SuperscriptBox["\[Pi]", "3"]]], "81"]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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