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http://functions.wolfram.com/10.04.20.0002.01
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D[RiemannSiegelZ[z], {z, 2}] == (1/16) RiemannSiegelZ[z]
(2 (PolyGamma[1, 1/4 - (I z)/2] - PolyGamma[1, 1/4 + (I z)/2]) -
(PolyGamma[1/4 - (I z)/2] + PolyGamma[1/4 + (I z)/2] - 2 Log[Pi])^2 +
(-((8 Derivative[1][Zeta][1/2 + I z])/Zeta[1/2 + I z]))
(PolyGamma[1/4 - (I z)/2] + PolyGamma[1/4 + (I z)/2] - 2 Log[Pi]) -
(16 Derivative[2][Zeta][1/2 + I z])/Zeta[1/2 + I z])
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Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "2"]], "}"]]], RowBox[List["RiemannSiegelZ", "[", "z", "]"]]]], "\[Equal]", RowBox[List[FractionBox["1", "16"], " ", RowBox[List["RiemannSiegelZ", "[", "z", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List[FractionBox["1", "4"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], "2"]]]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List[FractionBox["1", "4"], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], "2"]]]]], "]"]]]], ")"]]]], "-", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "4"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], "2"]]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "4"], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], "2"]]], "]"]], "-", RowBox[List["2", " ", RowBox[List["Log", "[", "\[Pi]", "]"]]]]]], ")"]], "2"], "+", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["8", RowBox[List[SuperscriptBox["Zeta", "\[Prime]", Rule[MultilineFunction, None]], "[", RowBox[List[FractionBox["1", "2"], "+", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "]"]]]], RowBox[List["Zeta", "[", RowBox[List[FractionBox["1", "2"], "+", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "]"]]]]], RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "4"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], "2"]]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "4"], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], "2"]]], "]"]], "-", RowBox[List["2", " ", RowBox[List["Log", "[", "\[Pi]", "]"]]]]]], ")"]]]], " ", "-", FractionBox[RowBox[List["16", " ", RowBox[List[SuperscriptBox["Zeta", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", RowBox[List[FractionBox["1", "2"], "+", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "]"]]]], RowBox[List["Zeta", "[", RowBox[List[FractionBox["1", "2"], "+", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "]"]]]]], ")"]]]]]]]]
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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> <mfrac> <mrow> <msup> <mo> ∂ </mo> <mn> 2 </mn> </msup> <mrow> <semantics> <mi> Z </mi> <annotation encoding='Mathematica'> TagBox["Z", RiemannSiegelZ] </annotation> </semantics> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> ∂ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 16 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <semantics> <mi> Z </mi> <annotation encoding='Mathematica'> TagBox["Z", RiemannSiegelZ] </annotation> </semantics> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <msup> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> + </mo> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> + </mo> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> π </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mfrac> <mrow> <mn> 8 </mn> <mtext> </mtext> </mrow> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "z"]], "+", FractionBox["1", "2"]]], Rule[Editable, True]], ")"]], InterpretTemplate[Function[$CellContext`e, Zeta[$CellContext`e]]]] </annotation> </semantics> </mfrac> <mo> ⁢ </mo> <mrow> <msup> <mi> ζ </mi> <mo> ′ </mo> </msup> <mo> ( </mo> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> + </mo> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> π </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mn> 16 </mn> <mtext> </mtext> </mrow> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "z"]], "+", FractionBox["1", "2"]]], Rule[Editable, True]], ")"]], InterpretTemplate[Function[$CellContext`e, Zeta[$CellContext`e]]]] </annotation> </semantics> </mfrac> <mo> ⁢ </mo> <mrow> <msup> <mi> ζ </mi> <mi> ′′ </mi> </msup> <mo> ( </mo> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <cn type='integer'> 2 </cn> </degree> </bvar> <apply> <ci> RiemannSiegelZ </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 16 </cn> <apply> <ci> RiemannSiegelZ </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <apply> <plus /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <apply> <plus /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <times /> <imaginaryi /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <plus /> <apply> <ci> PolyGamma </ci> <apply> <plus /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <times /> <imaginaryi /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ln /> <pi /> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <apply> <ci> Zeta </ci> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> D </ci> <apply> <ci> Zeta </ci> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <plus /> <apply> <ci> PolyGamma </ci> <apply> <plus /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <times /> <imaginaryi /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ln /> <pi /> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> 16 </cn> <apply> <power /> <apply> <ci> Zeta </ci> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> D </ci> <apply> <ci> Zeta </ci> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <list> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 2 </cn> </list> </apply> </apply> </apply> </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["z_", ",", "2"]], "}"]]]]], RowBox[List["RiemannSiegelZ", "[", "z_", "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "16"], " ", RowBox[List["RiemannSiegelZ", "[", "z", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List[FractionBox["1", "4"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], "2"]]]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List[FractionBox["1", "4"], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], "2"]]]]], "]"]]]], ")"]]]], "-", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "4"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], "2"]]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "4"], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], "2"]]], "]"]], "-", RowBox[List["2", " ", RowBox[List["Log", "[", "\[Pi]", "]"]]]]]], ")"]], "2"], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["8", " ", RowBox[List[SuperscriptBox["Zeta", "\[Prime]", Rule[MultilineFunction, None]], "[", RowBox[List[FractionBox["1", "2"], "+", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "4"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], "2"]]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "4"], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], "2"]]], "]"]], "-", RowBox[List["2", " ", RowBox[List["Log", "[", "\[Pi]", "]"]]]]]], ")"]]]], RowBox[List["Zeta", "[", RowBox[List[FractionBox["1", "2"], "+", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "]"]]], "-", FractionBox[RowBox[List["16", " ", RowBox[List[SuperscriptBox["Zeta", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", RowBox[List[FractionBox["1", "2"], "+", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "]"]]]], RowBox[List["Zeta", "[", RowBox[List[FractionBox["1", "2"], "+", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "]"]]]]], ")"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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