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http://functions.wolfram.com/10.09.21.0005.01
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Integrate[\[Nu]^(\[Alpha] - 1) PolyLog[\[Nu], p, z], \[Nu]] ==
(-\[Nu]^\[Alpha]) Sum[((-1)^k StirlingS1[k + p, p]
Gamma[\[Alpha], \[Nu] Log[k + p]] z^(k + p))/
((k + p)! (\[Nu] Log[k + p])^\[Alpha]), {k, 0, Infinity}]
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["\[Nu]", RowBox[List["\[Alpha]", "-", "1"]]], RowBox[List["PolyLog", "[", RowBox[List["\[Nu]", ",", "p", ",", "z"]], "]"]], RowBox[List["\[DifferentialD]", "\[Nu]"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", SuperscriptBox["\[Nu]", "\[Alpha]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["StirlingS1", "[", RowBox[List[RowBox[List["k", "+", "p"]], ",", "p"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List["\[Nu]", " ", RowBox[List["Log", "[", RowBox[List["k", "+", "p"]], "]"]]]]]], "]"]], SuperscriptBox["z", RowBox[List["k", "+", "p"]]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["k", "+", "p"]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[Nu]", " ", RowBox[List["Log", "[", RowBox[List["k", "+", "p"]], "]"]]]], ")"]], "\[Alpha]"]]], ")"]]]]]]]]]]]]
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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> <mo> ∫ </mo> <mrow> <mrow> <msup> <mi> ν </mi> <mrow> <mi> α </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <msub> <semantics> <mi> S </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mrow> <mi> ν </mi> <mo> , </mo> <mi> p </mi> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> ν </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> - </mo> <msup> <mi> ν </mi> <mi> α </mi> </msup> </mrow> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> ⁢ </mo> <msubsup> <semantics> <mi> S </mi> <annotation encoding='Mathematica'> TagBox["S", StirlingS1] </annotation> </semantics> <mrow> <mi> k </mi> <mo> + </mo> <mi> p </mi> </mrow> <mrow> <mo> ( </mo> <mi> p </mi> <mo> ) </mo> </mrow> </msubsup> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> α </mi> <mo> , </mo> <mrow> <mi> ν </mi> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mi> k </mi> <mo> + </mo> <mi> p </mi> </mrow> </msup> </mrow> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> α </mi> </msup> </mrow> </mfrac> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> ν </ci> </bvar> <apply> <times /> <apply> <power /> <ci> ν </ci> <apply> <plus /> <ci> α </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> PolyLog </ci> <ci> ν </ci> <ci> p </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> ν </ci> <ci> α </ci> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <ci> StirlingS1 </ci> <apply> <plus /> <ci> k </ci> <ci> p </ci> </apply> <ci> p </ci> </apply> <apply> <ci> Gamma </ci> <ci> α </ci> <apply> <times /> <ci> ν </ci> <apply> <ln /> <apply> <plus /> <ci> k </ci> <ci> p </ci> </apply> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> k </ci> <ci> p </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <apply> <plus /> <ci> k </ci> <ci> p </ci> </apply> </apply> <apply> <power /> <apply> <times /> <ci> ν </ci> <apply> <ln /> <apply> <plus /> <ci> k </ci> <ci> p </ci> </apply> </apply> </apply> <ci> α </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["\[Nu]_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", RowBox[List["PolyLog", "[", RowBox[List["\[Nu]_", ",", "p_", ",", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "\[Nu]_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", SuperscriptBox["\[Nu]", "\[Alpha]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["StirlingS1", "[", RowBox[List[RowBox[List["k", "+", "p"]], ",", "p"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List["\[Nu]", " ", RowBox[List["Log", "[", RowBox[List["k", "+", "p"]], "]"]]]]]], "]"]], " ", SuperscriptBox["z", RowBox[List["k", "+", "p"]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["k", "+", "p"]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[Nu]", " ", RowBox[List["Log", "[", RowBox[List["k", "+", "p"]], "]"]]]], ")"]], "\[Alpha]"]]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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