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http://functions.wolfram.com/10.01.21.0002.01
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Integrate[s^(\[Alpha] - 1) Zeta[s], s] == s^\[Alpha]/\[Alpha] -
s^\[Alpha] Sum[Gamma[\[Alpha], s Log[k]]/(s Log[k])^\[Alpha],
{k, 2, Infinity}]
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["s", RowBox[List["\[Alpha]", "-", "1"]]], RowBox[List["Zeta", "[", "s", "]"]], RowBox[List["\[DifferentialD]", "s"]]]]]], "\[Equal]", RowBox[List[FractionBox[SuperscriptBox["s", "\[Alpha]"], "\[Alpha]"], "-", RowBox[List[SuperscriptBox["s", "\[Alpha]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "2"]], "\[Infinity]"], FractionBox[RowBox[List[" ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List["s", " ", RowBox[List["Log", "[", "k", "]"]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["s", " ", RowBox[List["Log", "[", "k", "]"]]]], ")"]], "\[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> s </mi> <mrow> <mi> α </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox["s", Rule[Editable, True]], ")"]], InterpretTemplate[Function[$CellContext`e, Zeta[$CellContext`e]]]] </annotation> </semantics> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> s </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <msup> <mi> s </mi> <mi> α </mi> </msup> <mi> α </mi> </mfrac> <mo> - </mo> <mrow> <msup> <mi> s </mi> <mi> α </mi> </msup> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 2 </mn> </mrow> <mi> ∞ </mi> </munderover> <mfrac> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> α </mi> <mo> , </mo> <mrow> <mi> s </mi> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> s </mi> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> α </mi> </msup> </mfrac> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> s </ci> </bvar> <apply> <times /> <apply> <power /> <ci> s </ci> <apply> <plus /> <ci> α </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Zeta </ci> <ci> s </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> s </ci> <ci> α </ci> </apply> <apply> <power /> <ci> α </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> s </ci> <ci> α </ci> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 2 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <ci> Gamma </ci> <ci> α </ci> <apply> <times /> <ci> s </ci> <apply> <ln /> <ci> k </ci> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <times /> <ci> s </ci> <apply> <ln /> <ci> k </ci> </apply> </apply> <ci> α </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </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["s_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", RowBox[List["Zeta", "[", "s_", "]"]]]], RowBox[List["\[DifferentialD]", "s_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[SuperscriptBox["s", "\[Alpha]"], "\[Alpha]"], "-", RowBox[List[SuperscriptBox["s", "\[Alpha]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "2"]], "\[Infinity]"], FractionBox[RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List["s", " ", RowBox[List["Log", "[", "k", "]"]]]]]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["s", " ", RowBox[List["Log", "[", "k", "]"]]]], ")"]], "\[Alpha]"]]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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