Im[PolyLog[4, 2]] == (-(Pi/6)) Log[2]^3

Cell[BoxData[RowBox[List[RowBox[List["Im", "[", RowBox[List["PolyLog", "[", RowBox[List["4", ",", "2"]], "]"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["\[Pi]", "6"]]], SuperscriptBox[RowBox[List["Log", "[", "2", "]"]], "3"]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mn> 4 </mn> </msub> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mi> &#960; </mi> <mn> 6 </mn> </mfrac> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> log </mi> <mn> 3 </mn> </msup> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <imaginary /> <apply> <ci> PolyLog </ci> <cn type='integer'> 4 </cn> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 6 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Im", "[", RowBox[List["PolyLog", "[", RowBox[List["4", ",", "2"]], "]"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List["-", "\[Pi]"]], ")"]], " ", SuperscriptBox[RowBox[List["Log", "[", "2", "]"]], "3"]]]]]]]

2001-10-29