![](/common/images/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
http://functions.wolfram.com/10.07.21.0012.01
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
Integrate[t^\[Alpha] PolyLog[2, t], {t, 0, 1}] ==
(1/(6 \[Alpha] (1 + \[Alpha])^3))
(-6 + \[Alpha] (-12 + Pi^2 (1 + \[Alpha])^2) - 6 \[Alpha] (1 + \[Alpha])
(EulerGamma + PolyGamma[\[Alpha]])) /; Re[\[Alpha]] > -2
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", "1"], RowBox[List[SuperscriptBox["t", "\[Alpha]"], RowBox[List["PolyLog", "[", RowBox[List["2", ",", "t"]], "]"]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["6", " ", "\[Alpha]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "\[Alpha]"]], ")"]], "3"]]]], RowBox[List["(", RowBox[List[RowBox[List["-", "6"]], "+", RowBox[List["\[Alpha]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "12"]], "+", RowBox[List[SuperscriptBox["\[Pi]", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "\[Alpha]"]], ")"]], "2"]]]]], ")"]]]], "-", RowBox[List["6", " ", "\[Alpha]", " ", RowBox[List["(", RowBox[List["1", "+", "\[Alpha]"]], ")"]], " ", RowBox[List["(", RowBox[List["EulerGamma", "+", RowBox[List["PolyGamma", "[", "\[Alpha]", "]"]]]], ")"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["Re", "[", "\[Alpha]", "]"]], ">", RowBox[List["-", "2"]]]]]]]]
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msubsup> <mo> ∫ </mo> <mn> 0 </mn> <mn> 1 </mn> </msubsup> <mrow> <msup> <mi> t </mi> <mi> α </mi> </msup> <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> <mi> t </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mfrac> <mrow> <mrow> <mi> α </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> π </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> α </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mn> 12 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 6 </mn> <mo> ⁢ </mo> <mi> α </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> α </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mi> α </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <semantics> <mi> ℽ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubledGamma]", Function[EulerGamma]] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 6 </mn> </mrow> <mrow> <mn> 6 </mn> <mo> ⁢ </mo> <mi> α </mi> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> α </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mrow> </mfrac> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> α </mi> <mo> ) </mo> </mrow> <mo> > </mo> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <cn type='integer'> 1 </cn> </uplimit> <apply> <times /> <apply> <power /> <ci> t </ci> <ci> α </ci> </apply> <apply> <ci> PolyLog </ci> <cn type='integer'> 2 </cn> <ci> t </ci> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> α </ci> <apply> <plus /> <apply> <times /> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> α </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -12 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 6 </cn> <ci> α </ci> <apply> <plus /> <ci> α </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> PolyGamma </ci> <ci> ψ </ci> </apply> <ci> α </ci> </apply> <eulergamma /> </apply> </apply> </apply> <cn type='integer'> -6 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 6 </cn> <ci> α </ci> <apply> <power /> <apply> <plus /> <ci> α </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <gt /> <apply> <real /> <ci> α </ci> </apply> <cn type='integer'> -2 </cn> </apply> </apply> </annotation-xml> </semantics> </math>
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/clear.gif)
| ![](/images/home/spacer.gif)
| ![](/images/home/spacer.gif)
| ![](/images/home/spacer.gif)
| ![](/images/home/spacer.gif)
| | ![](/images/home/spacer.gif)
| ![](/images/home/spacer.gif)
| ![](/images/home/spacer.gif)
| ![](/images/home/spacer.gif)
| ![](/images/home/spacer.gif)
| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "1"], RowBox[List[RowBox[List[SuperscriptBox["t_", "\[Alpha]_"], " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", "t_"]], "]"]]]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["-", "6"]], "+", RowBox[List["\[Alpha]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "12"]], "+", RowBox[List[SuperscriptBox["\[Pi]", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "\[Alpha]"]], ")"]], "2"]]]]], ")"]]]], "-", RowBox[List["6", " ", "\[Alpha]", " ", RowBox[List["(", RowBox[List["1", "+", "\[Alpha]"]], ")"]], " ", RowBox[List["(", RowBox[List["EulerGamma", "+", RowBox[List["PolyGamma", "[", "\[Alpha]", "]"]]]], ")"]]]]]], RowBox[List["6", " ", "\[Alpha]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "\[Alpha]"]], ")"]], "3"]]]], "/;", RowBox[List[RowBox[List["Re", "[", "\[Alpha]", "]"]], ">", RowBox[List["-", "2"]]]]]]]]]] |
| ![](/images/home/spacer.gif)
| ![](/images/home/spacer.gif)
| ![](/images/home/spacer.gif)
| ![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
Date Added to functions.wolfram.com (modification date)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/common/images/spacer.gif) |
|
|