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http://functions.wolfram.com/10.02.20.0018.01
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Derivative[2, 0][Zeta][s, m] == Derivative[2][Zeta][s] -
Sum[Log[k + 1]^2/(k + 1)^s, {k, 0, m - 2}] /;
Element[m, Integers] && m >= -1
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["Zeta", TagBox[RowBox[List["(", RowBox[List["2", ",", "0"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List["s", ",", "m"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox["Zeta", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "s", "]"]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["m", "-", "2"]]], FractionBox[SuperscriptBox[RowBox[List["Log", "[", RowBox[List["k", "+", "1"]], "]"]], "2"], SuperscriptBox[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], "s"]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[GreaterEqual]", RowBox[List["-", "1"]]]]]]]]]]
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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> <mrow> <msup> <mi> ζ </mi> <semantics> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", RowBox[List["2", ",", "0"]], ")"]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mrow> <mi> s </mi> <mo> , </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <msup> <mi> ζ </mi> <mi> ′′ </mi> </msup> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> m </mi> <mo> - </mo> <mn> 2 </mn> </mrow> </munderover> <mfrac> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> s </mi> </msup> </mfrac> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> m </mi> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> m </mi> <mo> ≥ </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <apply> <partialdiff /> <list> <cn type='integer'> 2 </cn> <cn type='integer'> 0 </cn> </list> <ci> Zeta </ci> </apply> <ci> s </ci> <ci> m </ci> </apply> <apply> <plus /> <apply> <partialdiff /> <bvar> <ci> s </ci> <degree> <cn type='integer'> 2 </cn> </degree> </bvar> <apply> <ci> Zeta </ci> <ci> s </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> m </ci> <cn type='integer'> -2 </cn> </apply> </uplimit> <apply> <times /> <apply> <power /> <apply> <ln /> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <ci> s </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> m </ci> <integers /> </apply> <apply> <geq /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox["Zeta", TagBox[RowBox[List["(", RowBox[List["2", ",", "0"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List["s_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["Zeta", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "s", "]"]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["m", "-", "2"]]], FractionBox[SuperscriptBox[RowBox[List["Log", "[", RowBox[List["k", "+", "1"]], "]"]], "2"], SuperscriptBox[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], "s"]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", RowBox[List["-", "1"]]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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