Hurwitz zeta function: Integration (formula 10.02.21.0004)

Zeta

$Mathematica Notation$

Zeta Functions and Polylogarithms

Zeta[s,a]

Integration

Indefinite integration

Involving only one direct function with respect to a

For zeta^{^}(s,a)

http://functions.wolfram.com/10.02.21.0004.01

Input Form

Integrate[a^(\[Alpha] - 1) ZetaClassical[s, a], a] == (a^\[Alpha]/(\[Alpha] - s)) Sum[(1/((a + k)^s/((a + k)/a)^s)) Hypergeometric2F1[s - \[Alpha], s, 1 + s - \[Alpha], -(k/a)], {k, 0, Infinity}]

Standard Form

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MathML Form

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</mo> <mrow> <mi> s </mi> <mo> - </mo> <mi> α </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <mfrac> <mi> k </mi> <mi> a </mi> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["s", "-", "\[Alpha]"]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox["s", Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["s", "-", "\[Alpha]", "+", "1"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[RowBox[List["-", FractionBox["k", "a"]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]] </annotation> </semantics> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mo> ∫ </mo> <mrow> <mrow> <msup> <mi> a </mi> <mrow> <mi> α </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <semantics> <mrow> <mover> <mi> ζ </mi> <mo> ^ </mo> </mover> <mo> ( </mo> <mrow> <mi> s </mi> <mo> , </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[OverscriptBox["\[Zeta]", "^"], "(", RowBox[List["s", ",", TagBox["a", Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[List[$CellContext`e1, $CellContext`e2, $CellContext`e3], LerchPhi[$CellContext`e1, $CellContext`e2, $CellContext`e3]]]] </annotation> </semantics> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> a </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <msup> <mi> a </mi> <mi> α </mi> </msup> <mtext> </mtext> </mrow> <mrow> <mi> α </mi> <mo> - </mo> <mi> s </mi> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mfrac> <mn> 1 </mn> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <mi> s </mi> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mi> k </mi> </mrow> <mi> a </mi> </mfrac> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> s </mi> </mrow> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> s </mi> <mo> - </mo> <mi> α </mi> </mrow> <mo> , </mo> <mi> s </mi> </mrow> <mo> ; 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Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["a_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", RowBox[List["ZetaClassical", "[", RowBox[List["s_", ",", "a_"]], "]"]]]], RowBox[List["\[DifferentialD]", "a_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["a", "\[Alpha]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["s", "-", "\[Alpha]"]], ",", "s", ",", RowBox[List["1", "+", "s", "-", "\[Alpha]"]], ",", RowBox[List["-", FractionBox["k", "a"]]]]], "]"]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "k"]], ")"]], "s"], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["a", "+", "k"]], "a"], ")"]], RowBox[List["-", "s"]]]]]]]]]], RowBox[List["\[Alpha]", "-", "s"]]]]]]]

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

2001-10-29

Zeta[s]