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http://functions.wolfram.com/10.03.21.0002.01
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Integrate[RiemannSiegelTheta[z], z] == (-(1/4)) z^2 Log[Pi] -
PolyGamma[-2, 1/4 - (I z)/2] - PolyGamma[-2, 1/4 + (I z)/2]
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["RiemannSiegelTheta", "[", "z", "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", SuperscriptBox["z", "2"], " ", RowBox[List["Log", "[", "\[Pi]", "]"]]]], "-", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "2"]], ",", RowBox[List[FractionBox["1", "4"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], "2"]]]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "2"]], ",", RowBox[List[FractionBox["1", "4"], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], "2"]]]]], "]"]]]]]]]]
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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> <mo> ∫ </mo> <mrow> <mrow> <semantics> <mi> ϑ </mi> <annotation encoding='Mathematica'> TagBox["\[CurlyTheta]", RiemannSiegelTheta] </annotation> </semantics> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo>  </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> π </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <msup> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 4 </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> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <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> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <ci> RiemannSiegelTheta </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <ln /> <pi /> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <cn type='integer'> -2 </cn> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <cn type='integer'> -2 </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> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List["RiemannSiegelTheta", "[", "z_", "]"]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", SuperscriptBox["z", "2"], " ", RowBox[List["Log", "[", "\[Pi]", "]"]]]], "-", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "2"]], ",", RowBox[List[FractionBox["1", "4"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], "2"]]]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "2"]], ",", RowBox[List[FractionBox["1", "4"], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], "2"]]]]], "]"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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