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variants of this functions
PolyLog




Mathematica Notation

Traditional Notation









Zeta Functions and Polylogarithms > PolyLog[nu,z] > Integral representations > Multiple integral representations





http://functions.wolfram.com/10.08.07.0009.01









  


  








Input Form





PolyLog[2, z] == -Integrate[(1/w) Integrate[1/t, {t, 1, 1 - w}], {w, 0, z}]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["PolyLog", "[", RowBox[List["2", ",", "z"]], "]"]], "\[Equal]", RowBox[List["-", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "z"], RowBox[List[FractionBox["1", "w"], RowBox[List["(", RowBox[List[SubsuperscriptBox["\[Integral]", "1", RowBox[List["1", "-", "w"]]], RowBox[List[FractionBox["1", "t"], RowBox[List["\[DifferentialD]", "t"]]]]]], ")"]], RowBox[List["\[DifferentialD]", "w"]]]]]]]]]]]]








MathML Form







<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> 2 </mn> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mo> - </mo> <mrow> <msubsup> <mo> &#8747; </mo> <mn> 0 </mn> <mi> z </mi> </msubsup> <mrow> <mfrac> <mn> 1 </mn> <mi> w </mi> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msubsup> <mo> &#8747; </mo> <mn> 1 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> w </mi> </mrow> </msubsup> <mrow> <mfrac> <mn> 1 </mn> <mi> t </mi> </mfrac> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> w </mi> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> PolyLog </ci> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <int /> <bvar> <ci> w </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> z </ci> </uplimit> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> w </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> </uplimit> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> t </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["PolyLog", "[", RowBox[List["2", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["-", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "z"], RowBox[List[FractionBox[RowBox[List[SubsuperscriptBox["\[Integral]", "1", RowBox[List["1", "-", "w"]]], RowBox[List[FractionBox["1", "t"], RowBox[List["\[DifferentialD]", "t"]]]]]], "w"], RowBox[List["\[DifferentialD]", "w"]]]]]]]]]]]]


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





2001-10-29