Riemann zeta function: Differentiation (formula 10.01.20.0009)

Zeta

$Mathematica Notation$

Zeta Functions and Polylogarithms

Zeta[s]

Differentiation

Low-order differentiation

Derivatives at zero

http://functions.wolfram.com/10.01.20.0009.01

Input Form

Derivative[4][Zeta][0] == (3 EulerGamma^4)/2 - (19 Pi^4)/480 + 4 EulerGamma^3 Log[2 Pi] - (1/2) Log[2 Pi]^4 - (1/4) Pi^2 (Log[2 Pi]^2 - 2 StieltjesGamma[1]) + 6 Log[2 Pi]^2 StieltjesGamma[1] + EulerGamma^2 (Pi^2/4 + 3 Log[2 Pi]^2 + 6 StieltjesGamma[1]) + 6 Log[2 Pi] StieltjesGamma[2] + 6 EulerGamma (2 Log[2 Pi] StieltjesGamma[1] + StieltjesGamma[2]) + 2 StieltjesGamma[3] - 4 Log[2 Pi] Zeta[3]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[SuperscriptBox["Zeta", TagBox[RowBox[List["(", "4", ")"]], Derivative], Rule[MultilineFunction, None]], "[", "0", "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["3", " ", SuperscriptBox["EulerGamma", "4"]]], "2"], "-", FractionBox[RowBox[List["19", " ", SuperscriptBox["\[Pi]", "4"]]], "480"], "+", RowBox[List["4", " ", SuperscriptBox["EulerGamma", "3"], " ", RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]]]], "-", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox[RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]], "4"]]], "-", RowBox[List[FractionBox["1", "4"], " ", SuperscriptBox["\[Pi]", "2"], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]], "2"], "-", RowBox[List["2", " ", RowBox[List["StieltjesGamma", "[", "1", "]"]]]]]], ")"]]]], "+", RowBox[List["6", " ", SuperscriptBox[RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]], "2"], " ", RowBox[List["StieltjesGamma", "[", "1", "]"]]]], "+", RowBox[List[SuperscriptBox["EulerGamma", "2"], " ", RowBox[List["(", RowBox[List[FractionBox[SuperscriptBox["\[Pi]", "2"], "4"], "+", RowBox[List["3", " ", SuperscriptBox[RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]], "2"]]], "+", RowBox[List["6", " ", RowBox[List["StieltjesGamma", "[", "1", "]"]]]]]], ")"]]]], "+", RowBox[List["6", " ", RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]], " ", RowBox[List["StieltjesGamma", "[", "2", "]"]]]], "+", RowBox[List["6", " ", "EulerGamma", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]], " ", RowBox[List["StieltjesGamma", "[", "1", "]"]]]], "+", RowBox[List["StieltjesGamma", "[", "2", "]"]]]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["StieltjesGamma", "[", "3", "]"]]]], "-", RowBox[List["4", " ", RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]], " ", RowBox[List["Zeta", "[", "3", "]"]]]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msup> <mi> ζ </mi> <semantics> <mrow> <mo> ( </mo> <mn> 4 </mn> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", "4", ")"]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mn> 0 </mn> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> ⁢ </mo> <msup> <mi> π </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msub> <semantics> <mi> γ </mi> <annotation encoding='Mathematica'> TagBox["\[Gamma]", StieltjesGamma] </annotation> </semantics> <mn> 1 </mn> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 6 </mn> <mo> ⁢ </mo> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msub> <semantics> <mi> γ </mi> <annotation encoding='Mathematica'> TagBox["\[Gamma]", StieltjesGamma] </annotation> </semantics> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <mrow> <msup> <semantics> <mi> ℽ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] </annotation> </semantics> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 6 </mn> <mo> ⁢ </mo> <msub> <semantics> <mi> γ </mi> <annotation encoding='Mathematica'> TagBox["\[Gamma]", StieltjesGamma] </annotation> </semantics> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mfrac> <msup> <mi> π </mi> <mn> 2 </mn> </msup> <mn> 4 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 6 </mn> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msub> <semantics> <mi> γ </mi> <annotation encoding='Mathematica'> TagBox["\[Gamma]", StieltjesGamma] </annotation> </semantics> <mn> 2 </mn> </msub> </mrow> <mo> + </mo> <mrow> <mn> 6 </mn> <mo> ⁢ </mo> <semantics> <mi> ℽ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msub> <semantics> <mi> γ </mi> <annotation encoding='Mathematica'> TagBox["\[Gamma]", StieltjesGamma] </annotation> </semantics> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <msub> <semantics> <mi> γ </mi> <annotation encoding='Mathematica'> TagBox["\[Gamma]", StieltjesGamma] </annotation> </semantics> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msub> <semantics> <mi> γ </mi> <annotation encoding='Mathematica'> TagBox["\[Gamma]", StieltjesGamma] </annotation> </semantics> <mn> 3 </mn> </msub> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mn> 3 </mn> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox["3", Zeta, Rule[Editable, True]], ")"]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <msup> <mi> log </mi> <mn> 4 </mn> </msup> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <msup> <semantics> <mi> ℽ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] </annotation> </semantics> <mn> 3 </mn> </msup> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mfrac> <mrow> <mn> 19 </mn> <mo> ⁢ </mo> <msup> <mi> π </mi> <mn> 4 </mn> </msup> </mrow> <mn> 480 </mn> </mfrac> <mo> + </mo> <mfrac> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <msup> <semantics> <mi> ℽ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] </annotation> </semantics> <mn> 4 </mn> </msup> </mrow> <mn> 2 </mn> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> D </ci> <apply> <ci> Zeta </ci> <cn type='integer'> 0 </cn> </apply> <list> <cn type='integer'> 0 </cn> <cn type='integer'> 4 </cn> </list> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <power /> <apply> <ln /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> StieltjesGamma </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <power /> <apply> <ln /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ci> StieltjesGamma </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <eulergamma /> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <ci> StieltjesGamma </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <apply> <ln /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <ln /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> </apply> <apply> <ci> StieltjesGamma </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 6 </cn> <eulergamma /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ln /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> </apply> <apply> <ci> StieltjesGamma </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> StieltjesGamma </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> StieltjesGamma </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <ln /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> </apply> <apply> <ci> Zeta </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <power /> <apply> <ln /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> </apply> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <eulergamma /> <cn type='integer'> 3 </cn> </apply> <apply> <ln /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 19 </cn> <apply> <power /> <pi /> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <cn type='integer'> 480 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <eulergamma /> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox["Zeta", TagBox[RowBox[List["(", "4", ")"]], Derivative], Rule[MultilineFunction, None]], "[", "0", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["3", " ", SuperscriptBox["EulerGamma", "4"]]], "2"], "-", FractionBox[RowBox[List["19", " ", SuperscriptBox["\[Pi]", "4"]]], "480"], "+", RowBox[List["4", " ", SuperscriptBox["EulerGamma", "3"], " ", RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]]]], "-", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox[RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]], "4"]]], "-", RowBox[List[FractionBox["1", "4"], " ", SuperscriptBox["\[Pi]", "2"], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]], "2"], "-", RowBox[List["2", " ", RowBox[List["StieltjesGamma", "[", "1", "]"]]]]]], ")"]]]], "+", RowBox[List["6", " ", SuperscriptBox[RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]], "2"], " ", RowBox[List["StieltjesGamma", "[", "1", "]"]]]], "+", RowBox[List[SuperscriptBox["EulerGamma", "2"], " ", RowBox[List["(", RowBox[List[FractionBox[SuperscriptBox["\[Pi]", "2"], "4"], "+", RowBox[List["3", " ", SuperscriptBox[RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]], "2"]]], "+", RowBox[List["6", " ", RowBox[List["StieltjesGamma", "[", "1", "]"]]]]]], ")"]]]], "+", RowBox[List["6", " ", RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]], " ", RowBox[List["StieltjesGamma", "[", "2", "]"]]]], "+", RowBox[List["6", " ", "EulerGamma", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]], " ", RowBox[List["StieltjesGamma", "[", "1", "]"]]]], "+", RowBox[List["StieltjesGamma", "[", "2", "]"]]]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["StieltjesGamma", "[", "3", "]"]]]], "-", RowBox[List["4", " ", RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]], " ", RowBox[List["Zeta", "[", "3", "]"]]]]]]]]]]

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

2007-05-02

Zeta[s,a]