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RiemannSiegelZ




Mathematica Notation

Traditional Notation









Zeta Functions and Polylogarithms > RiemannSiegelZ[z] > Differentiation > Low-order differentiation





http://functions.wolfram.com/10.04.20.0001.01









  


  








Input Form





D[RiemannSiegelZ[z], z] == RiemannSiegelZ[z] ((I/4) (PolyGamma[1/4 - (I z)/2] + PolyGamma[1/4 + (I z)/2] - 2 Log[Pi]) + (1/Zeta[I z + 1/2]) Derivative[1][Zeta][1/2 + I z])








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", "z"], RowBox[List["RiemannSiegelZ", "[", "z", "]"]]]], "\[Equal]", RowBox[List[RowBox[List["RiemannSiegelZ", "[", "z", "]"]], RowBox[List["(", RowBox[List[RowBox[List[FractionBox["\[ImaginaryI]", "4"], " ", 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[FractionBox["1", RowBox[List["Zeta", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "z"]], "+", FractionBox["1", "2"]]], "]"]]], RowBox[List[SuperscriptBox["Zeta", "\[Prime]", Rule[MultilineFunction, None]], "[", RowBox[List[FractionBox["1", "2"], "+", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "]"]]]]]], ")"]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mfrac> <mrow> <mo> &#8706; </mo> <mrow> <semantics> <mi> Z </mi> <annotation encoding='Mathematica'> TagBox[&quot;Z&quot;, RiemannSiegelZ] </annotation> </semantics> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> &#8706; </mo> <mi> z </mi> </mrow> </mfrac> <mo> &#10869; </mo> <mrow> <mrow> <semantics> <mi> Z </mi> <annotation encoding='Mathematica'> TagBox[&quot;Z&quot;, RiemannSiegelZ] </annotation> </semantics> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mi> &#8520; </mi> <mn> 4 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#960; </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </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> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mfrac> <mrow> <msup> <mi> &#950; </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </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[&quot;\[Zeta]&quot;, &quot;(&quot;, TagBox[RowBox[List[RowBox[List[&quot;\[ImaginaryI]&quot;, &quot; &quot;, &quot;z&quot;]], &quot;+&quot;, FractionBox[&quot;1&quot;, &quot;2&quot;]]], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[$CellContext`e, Zeta[$CellContext`e]]]] </annotation> </semantics> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> z </ci> </bvar> <apply> <ci> RiemannSiegelZ </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <ci> RiemannSiegelZ </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <apply> <ln /> <pi /> </apply> </apply> <apply> <ci> PolyGamma </ci> <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> <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> </apply> <apply> <times /> <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> <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> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["z_"]]], RowBox[List["RiemannSiegelZ", "[", "z_", "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["RiemannSiegelZ", "[", "z", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", "\[ImaginaryI]", " ", 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[SuperscriptBox["Zeta", "\[Prime]", Rule[MultilineFunction, None]], "[", RowBox[List[FractionBox["1", "2"], "+", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "]"]], RowBox[List["Zeta", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "z"]], "+", FractionBox["1", "2"]]], "]"]]]]], ")"]]]]]]]]


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





2001-10-29