Polylogarithm: Identities (formula 10.08.17.0007)

PolyLog

$Mathematica Notation$

Zeta Functions and Polylogarithms

PolyLog[nu,z]

Identities

Functional identities

General cases

Involving two polyilogarithms

http://functions.wolfram.com/10.08.17.0007.01

Input Form

PolyLog[\[Nu], z] == (-E^(Pi I \[Nu])) PolyLog[\[Nu], 1/z] + E^((Pi I \[Nu])/2) ((2 Pi)^\[Nu]/Gamma[\[Nu]]) ZetaClassical[1 - \[Nu], Log[-z]/(2 Pi I) + 1/2] /; !IntervalMemberQ[Interval[{0, 1}], z]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["PolyLog", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "\[Nu]"]]]]], " ", RowBox[List["PolyLog", "[", RowBox[List["\[Nu]", ",", FractionBox["1", "z"]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "\[Nu]"]], "2"]], FractionBox[RowBox[List[" ", SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "\[Pi]"]], ")"]], "\[Nu]"], " "]], RowBox[List["Gamma", "[", "\[Nu]", "]"]]], RowBox[List["ZetaClassical", "[", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", RowBox[List[FractionBox[RowBox[List["Log", "[", RowBox[List["-", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]"]]], "+", FractionBox["1", "2"]]]]], "]"]]]]]]]], "/;", RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List["0", ",", "1"]], "}"]], "]"]], ",", " ", "z"]], "]"]], "]"]]]]]]

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> <mi> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mfrac> <mrow> <mtext> </mtext> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mi> ν </mi> </msup> </mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mn> 2 </mn> </mfrac> </msup> <mo> ⁢ </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> ν </mi> </mrow> <mo> , </mo> <mrow> <mfrac> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> </mfrac> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", RowBox[List[TagBox[RowBox[List["1", "-", "\[Nu]"]], Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox[RowBox[List["log", "(", RowBox[List["-", "z"]], ")"]], RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]"]]], "+", FractionBox["1", "2"]]], Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[List[$CellContext`e1, $CellContext`e2], Zeta[$CellContext`e1, $CellContext`e2]]]] </annotation> </semantics> </mrow> <mo> - </mo> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mi> ν </mi> </msub> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> z </mi> <mo> ∉ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mi> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mfrac> <mrow> <mtext> </mtext> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mi> ν </mi> </msup> </mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mn> 2 </mn> </mfrac> </msup> <mo> ⁢ </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> ν </mi> </mrow> <mo> , </mo> <mrow> <mfrac> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> </mfrac> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", RowBox[List[TagBox[RowBox[List["1", "-", "\[Nu]"]], Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox[RowBox[List["log", "(", RowBox[List["-", "z"]], ")"]], RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]"]]], "+", FractionBox["1", "2"]]], Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[List[$CellContext`e1, $CellContext`e2], Zeta[$CellContext`e1, $CellContext`e2]]]] </annotation> </semantics> </mrow> <mo> - </mo> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mi> ν </mi> </msub> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> z </mi> <mo> ∉ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["PolyLog", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "\[Nu]"]]]]], " ", RowBox[List["PolyLog", "[", RowBox[List["\[Nu]", ",", FractionBox["1", "z"]]], "]"]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "\[Nu]"]], "2"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "\[Pi]"]], ")"]], "\[Nu]"], " ", RowBox[List["ZetaClassical", "[", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", RowBox[List[FractionBox[RowBox[List["Log", "[", RowBox[List["-", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]"]]], "+", FractionBox["1", "2"]]]]], "]"]]]], RowBox[List["Gamma", "[", "\[Nu]", "]"]]]]], "/;", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List["0", ",", "1"]], "}"]], "]"]], ",", "z"]], "]"]]]]]]]]]]

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

2001-10-29

PolyLog[nu,p,z]
PolyLog[2,z]