Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











variants of this functions
PolyLog






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/10.07.21.0019.01









  


  










Input Form





Integrate[(Log[1 + a t] PolyLog[2, -(z/t^2)])/t, {t, 0, Infinity}] == (1/(480 a)) (120 Pi Sqrt[1/z] LerchPhi[-(1/(a^2 z)), 3, 1/2] - a (53 Pi^4 + 50 Pi^2 Log[1/(a^2 z)]^2 + 5 Log[1/(a^2 z)]^4 + 120 Log[1/(a^2 z)] PolyLog[3, -(1/(a^2 z))] - 360 PolyLog[4, -(1/(a^2 z))]))










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Log", "[", RowBox[List["1", "+", RowBox[List["a", " ", "t"]]]], "]"]], " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["-", FractionBox["z", SuperscriptBox["t", "2"]]]]]], "]"]]]], "t"], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["480", " ", "a"]]], RowBox[List["(", RowBox[List[RowBox[List["120", " ", "\[Pi]", " ", SqrtBox[FractionBox["1", "z"]], " ", RowBox[List["LerchPhi", "[", RowBox[List[RowBox[List["-", FractionBox["1", RowBox[List[SuperscriptBox["a", "2"], " ", "z"]]]]], ",", "3", ",", FractionBox["1", "2"]]], "]"]]]], "-", RowBox[List["a", " ", RowBox[List["(", RowBox[List[RowBox[List["53", " ", SuperscriptBox["\[Pi]", "4"]]], "+", RowBox[List["50", " ", SuperscriptBox["\[Pi]", "2"], " ", SuperscriptBox[RowBox[List["Log", "[", FractionBox["1", RowBox[List[SuperscriptBox["a", "2"], " ", "z"]]], "]"]], "2"]]], "+", RowBox[List["5", " ", SuperscriptBox[RowBox[List["Log", "[", FractionBox["1", RowBox[List[SuperscriptBox["a", "2"], " ", "z"]]], "]"]], "4"]]], "+", RowBox[List["120", " ", RowBox[List["Log", "[", FractionBox["1", RowBox[List[SuperscriptBox["a", "2"], " ", "z"]]], "]"]], " ", RowBox[List["PolyLog", "[", RowBox[List["3", ",", RowBox[List["-", FractionBox["1", RowBox[List[SuperscriptBox["a", "2"], " ", "z"]]]]]]], "]"]]]], "-", RowBox[List["360", " ", RowBox[List["PolyLog", "[", RowBox[List["4", ",", RowBox[List["-", FractionBox["1", RowBox[List[SuperscriptBox["a", "2"], " ", "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> 0 </mn> <mi> &#8734; </mi> </msubsup> <mrow> <mfrac> <mn> 1 </mn> <mi> t </mi> </mfrac> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> t </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <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> <mfrac> <mi> z </mi> <msup> <mi> t </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 480 </mn> <mo> &#8290; </mo> <mi> a </mi> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 120 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <msqrt> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> </msqrt> <mo> &#8290; </mo> <semantics> <mrow> <mi> &#934; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mfrac> </mrow> <mo> , </mo> <mn> 3 </mn> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[CapitalPhi]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[RowBox[List[&quot;-&quot;, FractionBox[&quot;1&quot;, RowBox[List[SuperscriptBox[&quot;a&quot;, &quot;2&quot;], &quot; &quot;, &quot;z&quot;]]]]], Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;3&quot;, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;1&quot;, &quot;2&quot;], Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[List[$CellContext`e1, $CellContext`e2, $CellContext`e3], LerchPhi[$CellContext`e1, $CellContext`e2, $CellContext`e3]]]] </annotation> </semantics> </mrow> <mo> - </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 5 </mn> <mo> &#8290; </mo> <mrow> <msup> <mi> log </mi> <mn> 4 </mn> </msup> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 50 </mn> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 120 </mn> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 53 </mn> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 360 </mn> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mn> 4 </mn> </msub> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> t </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <ci> a </ci> <ci> t </ci> </apply> </apply> </apply> <apply> <ci> PolyLog </ci> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <power /> <ci> t </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 480 </cn> <ci> a </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 120 </cn> <pi /> <apply> <power /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> LerchPhi </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> a </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 5 </cn> <apply> <power /> <apply> <ln /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 50 </cn> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ln /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 120 </cn> <apply> <ci> PolyLog </ci> <cn type='integer'> 3 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <ln /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 53 </cn> <apply> <power /> <pi /> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 360 </cn> <apply> <ci> PolyLog </ci> <cn type='integer'> 4 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Log", "[", RowBox[List["1", "+", RowBox[List["a_", " ", "t_"]]]], "]"]], " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["-", FractionBox["z_", SuperscriptBox["t_", "2"]]]]]], "]"]]]], "t_"], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["120", " ", "\[Pi]", " ", SqrtBox[FractionBox["1", "z"]], " ", RowBox[List["LerchPhi", "[", RowBox[List[RowBox[List["-", FractionBox["1", RowBox[List[SuperscriptBox["a", "2"], " ", "z"]]]]], ",", "3", ",", FractionBox["1", "2"]]], "]"]]]], "-", RowBox[List["a", " ", RowBox[List["(", RowBox[List[RowBox[List["53", " ", SuperscriptBox["\[Pi]", "4"]]], "+", RowBox[List["50", " ", SuperscriptBox["\[Pi]", "2"], " ", SuperscriptBox[RowBox[List["Log", "[", FractionBox["1", RowBox[List[SuperscriptBox["a", "2"], " ", "z"]]], "]"]], "2"]]], "+", RowBox[List["5", " ", SuperscriptBox[RowBox[List["Log", "[", FractionBox["1", RowBox[List[SuperscriptBox["a", "2"], " ", "z"]]], "]"]], "4"]]], "+", RowBox[List["120", " ", RowBox[List["Log", "[", FractionBox["1", RowBox[List[SuperscriptBox["a", "2"], " ", "z"]]], "]"]], " ", RowBox[List["PolyLog", "[", RowBox[List["3", ",", RowBox[List["-", FractionBox["1", RowBox[List[SuperscriptBox["a", "2"], " ", "z"]]]]]]], "]"]]]], "-", RowBox[List["360", " ", RowBox[List["PolyLog", "[", RowBox[List["4", ",", RowBox[List["-", FractionBox["1", RowBox[List[SuperscriptBox["a", "2"], " ", "z"]]]]]]], "]"]]]]]], ")"]]]]]], RowBox[List["480", " ", "a"]]]]]]]










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





2001-10-29





© 1998-2014 Wolfram Research, Inc.