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variants of this functions
PolyLog






Mathematica Notation

Traditional Notation









Zeta Functions and Polylogarithms > PolyLog[nu,z] > Identities > Functional identities > General cases > Involving two polyilogarithms





http://functions.wolfram.com/10.08.17.0010.01









  


  










Input Form





PolyLog[n, z] == (-1)^(n - 1) PolyLog[n, 1/z] - Log[-z]^n/n! + 2 Sum[(PolyLog[2 k, -1]/(n - 2 k)!) Log[-z]^(n - 2 k), {k, 1, Floor[n/2]}] /; !IntervalMemberQ[Interval[{0, 1}], z] && Element[n, Integers] && n > 0










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["PolyLog", "[", RowBox[List["n", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "1"]]], " ", RowBox[List["PolyLog", "[", RowBox[List["n", ",", FractionBox["1", "z"]]], "]"]]]], "-", FractionBox[SuperscriptBox[RowBox[List["Log", "[", RowBox[List["-", "z"]], "]"]], "n"], RowBox[List["n", "!"]]], "+", RowBox[List["2", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]], RowBox[List[FractionBox[RowBox[List[" ", RowBox[List["PolyLog", "[", RowBox[List[RowBox[List["2", " ", "k"]], ",", RowBox[List["-", "1"]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", RowBox[List["2", " ", "k"]]]], ")"]], "!"]]], SuperscriptBox[RowBox[List["Log", "[", RowBox[List["-", "z"]], "]"]], RowBox[List["n", "-", RowBox[List["2", " ", "k"]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List["0", ",", "1"]], "}"]], "]"]], ",", " ", "z"]], "]"]], "]"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]]]]]]]]










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> n </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <msup> <mi> log </mi> <mi> n </mi> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> </mfrac> </mrow> <mo> + </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mi> n </mi> </msub> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mo> &#8970; </mo> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> <mo> &#8971; </mo> </mrow> </munderover> <mrow> <mfrac> <mrow> <mtext> </mtext> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> </msub> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> log </mi> <mrow> <mi> n </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> </mrow> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> z </mi> <mo> &#8713; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </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> n </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <msup> <mi> log </mi> <mi> n </mi> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> </mfrac> </mrow> <mo> + </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mi> n </mi> </msub> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mo> &#8970; </mo> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> <mo> &#8971; </mo> </mrow> </munderover> <mrow> <mfrac> <mrow> <mtext> </mtext> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> </msub> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> log </mi> <mrow> <mi> n </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> </mrow> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> z </mi> <mo> &#8713; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["PolyLog", "[", RowBox[List["n_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "1"]]], " ", RowBox[List["PolyLog", "[", RowBox[List["n", ",", FractionBox["1", "z"]]], "]"]]]], "-", FractionBox[SuperscriptBox[RowBox[List["Log", "[", RowBox[List["-", "z"]], "]"]], "n"], RowBox[List["n", "!"]]], "+", RowBox[List["2", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]], FractionBox[RowBox[List[RowBox[List["PolyLog", "[", RowBox[List[RowBox[List["2", " ", "k"]], ",", RowBox[List["-", "1"]]]], "]"]], " ", SuperscriptBox[RowBox[List["Log", "[", RowBox[List["-", "z"]], "]"]], RowBox[List["n", "-", RowBox[List["2", " ", "k"]]]]]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", RowBox[List["2", " ", "k"]]]], ")"]], "!"]]]]]]]]], "/;", RowBox[List[RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List["0", ",", "1"]], "}"]], "]"]], ",", "z"]], "]"]]]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]










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





2001-10-29





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