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variants of this functions
PolyLog




Mathematica Notation

Traditional Notation









Zeta Functions and Polylogarithms > PolyLog[2,z] > Integration > Definite integration > For the direct function itself





http://functions.wolfram.com/10.07.21.0017.01









  


  








Input Form





Integrate[PolyLog[2, (-t) z]/(Sqrt[t] (1 + t)), {t, 1, Infinity}] == 4 Pi PolyLog[2, -Sqrt[z]]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "1", InterpretationBox["\[Infinity]", DirectedInfinity[1]]], RowBox[List[FractionBox[RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List[RowBox[List["-", "t"]], " ", "z"]]]], "]"]], RowBox[List[SqrtBox["t"], " ", RowBox[List["(", RowBox[List["1", "+", "t"]], ")"]]]]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", RowBox[List["4", "\[Pi]", " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["-", " ", SqrtBox["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> <msubsup> <mo> &#8747; </mo> <mn> 1 </mn> <semantics> <mi> &#8734; </mi> <annotation-xml encoding='MathML-Content'> <infinity /> </annotation-xml> </semantics> </msubsup> <mrow> <mfrac> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mn> 2 </mn> </msub> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> t </mi> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <msqrt> <mi> t </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> t </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mn> 2 </mn> </msub> <mo> ( </mo> <mrow> <mo> - </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <apply> <ci> DirectedInfinity </ci> <cn type='integer'> 1 </cn> </apply> </uplimit> <apply> <times /> <apply> <ci> Subscript </ci> <ci> PolyLog </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> t </ci> </apply> <ci> z </ci> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <ci> t </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <ci> t </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <pi /> <apply> <ci> PolyLog </ci> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", "1", "\[Infinity]"], RowBox[List[FractionBox[RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List[RowBox[List["-", "t_"]], " ", "z_"]]]], "]"]], RowBox[List[SqrtBox["t_"], " ", RowBox[List["(", RowBox[List["1", "+", "t_"]], ")"]]]]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["4", " ", "\[Pi]", " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["-", SqrtBox["z"]]]]], "]"]]]]]]]]


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





2001-10-29