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http://functions.wolfram.com/10.07.06.0023.01
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PolyLog[2, z] == Subscript[F, Infinity][z] /;
Subscript[F, n][z] == (-(1/2)) Log[-z]^2 - Pi^2/6 -
Sum[1/(k^2 z^k), {k, 1, n}] == (-(1/2)) Log[-z]^2 - Pi^2/6 +
z^(-1 - n) LerchPhi[1/z, 2, 1 + n] - PolyLog[2, 1/z] &&
Element[n, Integers] && n > 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["PolyLog", "[", RowBox[List["2", ",", "z"]], "]"]], "\[Equal]", RowBox[List[SubscriptBox["F", "\[Infinity]"], "[", "z", "]"]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["F", "n"], "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", SuperscriptBox[RowBox[List["Log", "[", RowBox[List["-", "z"]], "]"]], "2"]]], "-", FractionBox[SuperscriptBox["\[Pi]", "2"], "6"], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], FractionBox["1", RowBox[List[SuperscriptBox["k", "2"], SuperscriptBox["z", "k"]]]]]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", SuperscriptBox[RowBox[List["Log", "[", RowBox[List["-", "z"]], "]"]], "2"]]], "-", FractionBox[SuperscriptBox["\[Pi]", "2"], "6"], "+", RowBox[List[SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", RowBox[List["LerchPhi", "[", RowBox[List[FractionBox["1", "z"], ",", "2", ",", RowBox[List["1", "+", "n"]]]], "]"]]]], "-", RowBox[List["PolyLog", "[", RowBox[List["2", ",", FractionBox["1", "z"]]], "]"]]]]]], StyleBox[")", Rule[FontWeight, "Plain"]]]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]]]], ")"]]]]]]
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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> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mn> 2 </mn> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <msub> <mi> F </mi> <mi> ∞ </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> F </mi> <mi> n </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mfrac> <msup> <mi> π </mi> <mn> 2 </mn> </msup> <mn> 6 </mn> </mfrac> <mo> - </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mfrac> <mn> 1 </mn> <mrow> <msup> <mi> k </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> k </mi> </msup> </mrow> </mfrac> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mfrac> <msup> <mi> π </mi> <mn> 2 </mn> </msup> <mn> 6 </mn> </mfrac> <mo> + </mo> <mrow> <semantics> <mrow> <mi> Φ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> <mo> , </mo> <mn> 2 </mn> <mo> , </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[CapitalPhi]", "(", RowBox[List[TagBox[FractionBox["1", "z"], Rule[Editable, True]], ",", TagBox["2", Rule[Editable, True]], ",", TagBox[RowBox[List["n", "+", "1"]], Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[List[ZetaDump`e1$, ZetaDump`e2$, ZetaDump`e3$], LerchPhi[ZetaDump`e1$, ZetaDump`e2$, ZetaDump`e3$]]]] </annotation> </semantics> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> <mo> - </mo> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mn> 2 </mn> </msub> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ∧ </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <semantics> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> <annotation encoding='Mathematica'> TagBox[SuperscriptBox["\[DoubleStruckCapitalN]", "+"], Function[Integers]] </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> PolyLog </ci> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <apply> <apply> <ci> Subscript </ci> <ci> F </ci> <infinity /> </apply> <ci> z </ci> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <apply> <ci> Subscript </ci> <ci> F </ci> <ci> n </ci> </apply> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <ln /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 6 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <apply> <power /> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <ci> k </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <ln /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 6 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <ci> LerchPhi </ci> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyLog </ci> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["PolyLog", "[", RowBox[List["2", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SubscriptBox["F", "\[Infinity]"], "[", "z", "]"]], "/;", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["F", "n"], "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", SuperscriptBox[RowBox[List["Log", "[", RowBox[List["-", "z"]], "]"]], "2"]]], "-", FractionBox[SuperscriptBox["\[Pi]", "2"], "6"], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], FractionBox["1", RowBox[List[SuperscriptBox["k", "2"], " ", SuperscriptBox["z", "k"]]]]]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", SuperscriptBox[RowBox[List["Log", "[", RowBox[List["-", "z"]], "]"]], "2"]]], "-", FractionBox[SuperscriptBox["\[Pi]", "2"], "6"], "+", RowBox[List[SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", RowBox[List["LerchPhi", "[", RowBox[List[FractionBox["1", "z"], ",", "2", ",", RowBox[List["1", "+", "n"]]]], "]"]]]], "-", RowBox[List["PolyLog", "[", RowBox[List["2", ",", FractionBox["1", "z"]]], "]"]]]]]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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