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http://functions.wolfram.com/10.02.20.0062.01
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D[ZetaClassical[s, a + 1], {s, n}] == D[ZetaClassical[s, a], {s, n}] +
((-1)^(n - 1) Log[a]^n)/a^s /; Element[n, Integers] && n >= 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["s", ",", "n"]], "}"]]], RowBox[List["ZetaClassical", "[", RowBox[List["s", ",", RowBox[List["a", "+", "1"]]]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["s", ",", "n"]], "}"]]], RowBox[List["ZetaClassical", "[", RowBox[List["s", ",", "a"]], "]"]]]], "+", FractionBox[RowBox[List[" ", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "1"]]], SuperscriptBox[RowBox[List["Log", "[", "a", "]"]], "n"]]]]], SuperscriptBox["a", "s"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "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> n </mi> </msup> <semantics> <mrow> <mover> <mi> ζ </mi> <mo> ^ </mo> </mover> <mo> ( </mo> <mrow> <mi> s </mi> <mo> , </mo> <mrow> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[OverscriptBox["\[Zeta]", "^"], "(", RowBox[List[TagBox["s", Rule[Editable, True]], ",", TagBox[RowBox[List["a", "+", "1"]], Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[List[$CellContext`e1, $CellContext`e2], Zeta[$CellContext`e1, $CellContext`e2]]]] </annotation> </semantics> </mrow> <mrow> <mo> ∂ </mo> <msup> <mi> s </mi> <mi> n </mi> </msup> </mrow> </mfrac> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <msup> <mo> ∂ </mo> <mi> n </mi> </msup> <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[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> s </mi> <mi> n </mi> </msup> </mrow> </mfrac> <mo> + </mo> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <msup> <mi> log </mi> <mi> n </mi> </msup> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mrow> <msup> <mi> a </mi> <mi> s </mi> </msup> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> s </ci> <degree> <ci> n </ci> </degree> </bvar> <apply> <ci> Zeta </ci> <ci> s </ci> <apply> <plus /> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <partialdiff /> <bvar> <ci> s </ci> <degree> <ci> n </ci> </degree> </bvar> <apply> <ci> Zeta </ci> <ci> s </ci> <ci> a </ci> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <ln /> <ci> a </ci> </apply> <ci> n </ci> </apply> <apply> <power /> <apply> <power /> <ci> a </ci> <ci> s </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <ci> ℕ </ci> </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["s_", ",", "n_"]], "}"]]]]], RowBox[List["ZetaClassical", "[", RowBox[List["s_", ",", RowBox[List["a_", "+", "1"]]]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["s", ",", "n"]], "}"]]]]], RowBox[List["ZetaClassical", "[", RowBox[List["s", ",", "a"]], "]"]]]], "+", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "1"]]], " ", SuperscriptBox[RowBox[List["Log", "[", "a", "]"]], "n"]]], SuperscriptBox["a", "s"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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