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http://functions.wolfram.com/10.02.20.0010.01
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D[Zeta[s, a], {a, \[Alpha]}] == (Gamma[1 - s] a^(-s - \[Alpha]))/
Gamma[1 - s - \[Alpha]] +
Sum[Hypergeometric2F1Regularized[1, s, 1 - \[Alpha], -(a/k)]/k^s,
{k, 1, Infinity}]/a^\[Alpha] /; Re[s] > 1 && Re[a] > 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["a", ",", "\[Alpha]"]], "}"]]], RowBox[List["Zeta", "[", RowBox[List["s", ",", "a"]], "]"]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "-", "s"]], "]"]], SuperscriptBox["a", RowBox[List[RowBox[List["-", "s"]], "-", "\[Alpha]"]]]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "s", "-", "\[Alpha]"]], "]"]]], "+", RowBox[List[SuperscriptBox["a", RowBox[List["-", "\[Alpha]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[SuperscriptBox["k", RowBox[List["-", "s"]]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["1", ",", "s", ",", RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List["-", FractionBox["a", "k"]]]]], "]"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "s", "]"]], ">", "1"]], "\[And]", RowBox[List[RowBox[List["Re", "[", "a", "]"]], ">", "0"]]]]]]]]
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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> <mfrac> <mrow> <msup> <mo> ∂ </mo> <mi> α </mi> </msup> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> s </mi> <mo> , </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", RowBox[List[TagBox["s", Rule[Editable, True]], ",", TagBox["a", Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[List[$CellContext`e1, $CellContext`e2], Zeta[$CellContext`e1, $CellContext`e2]]]] </annotation> </semantics> </mrow> <mrow> <mo> ∂ </mo> <msup> <mi> a </mi> <mi> α </mi> </msup> </mrow> </mfrac> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> s </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> a </mi> <mrow> <mrow> <mo> - </mo> <mi> s </mi> </mrow> <mo> - </mo> <mi> α </mi> </mrow> </msup> </mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> s </mi> <mo> - </mo> <mi> α </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> + </mo> <mtext> </mtext> <mrow> <msup> <mi> a </mi> <mrow> <mo> - </mo> <mi> α </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <msup> <mi> k </mi> <mrow> <mo> - </mo> <mi> s </mi> </mrow> </msup> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 2 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> , </mo> <mi> s </mi> </mrow> <mo> ; </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> α </mi> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <mfrac> <mi> a </mi> <mi> k </mi> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox["s", Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["1", "-", "\[Alpha]"]], Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[RowBox[List["-", FractionBox["a", "k"]]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] </annotation> </semantics> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> <mo> > </mo> <mn> 1 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> > </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> a </ci> <degree> <ci> α </ci> </degree> </bvar> <apply> <ci> Zeta </ci> <ci> s </ci> <ci> a </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> </apply> <apply> <power /> <ci> a </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> <apply> <ci> Hypergeometric2F1Regularized </ci> <cn type='integer'> 1 </cn> <ci> s </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <gt /> <apply> <real /> <ci> s </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <gt /> <apply> <real /> <ci> a </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["a_", ",", "\[Alpha]_"]], "}"]]]]], RowBox[List["Zeta", "[", RowBox[List["s_", ",", "a_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "-", "s"]], "]"]], " ", SuperscriptBox["a", RowBox[List[RowBox[List["-", "s"]], "-", "\[Alpha]"]]]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "s", "-", "\[Alpha]"]], "]"]]], "+", RowBox[List[SuperscriptBox["a", RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[SuperscriptBox["k", RowBox[List["-", "s"]]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["1", ",", "s", ",", RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List["-", FractionBox["a", "k"]]]]], "]"]]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "s", "]"]], ">", "1"]], "&&", RowBox[List[RowBox[List["Re", "[", "a", "]"]], ">", "0"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
