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RamanujanTauZ






Mathematica Notation

Traditional Notation









Zeta Functions and Polylogarithms > RamanujanTauZ[z] > Primary definition





http://functions.wolfram.com/10.11.02.0001.01









  


  










Input Form





RamanujanTauZ[z] == (Pi^(-(1/2) - I z) RamanujanTauL[6 + I z] Gamma[6 + I z] Sqrt[Sinh[Pi z]/(z (1 + z^2) (4 + z^2) (9 + z^2) (16 + z^2) (25 + z^2))])/ 2^(I z)










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["RamanujanTauZ", "[", "z", "]"]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]]], " ", SuperscriptBox["\[Pi]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", RowBox[List["\[ImaginaryI]", " ", "z"]]]]], " ", RowBox[List["RamanujanTauL", "[", RowBox[List["6", "+", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["6", "+", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "]"]], SqrtBox[FractionBox[RowBox[List["Sinh", "[", RowBox[List["\[Pi]", " ", "z"]], "]"]], RowBox[List["z", " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["z", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List["4", "+", SuperscriptBox["z", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List["9", "+", SuperscriptBox["z", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List["16", "+", SuperscriptBox["z", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List["25", "+", SuperscriptBox["z", "2"]]], ")"]]]]]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> &#964;Z </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <msup> <mn> 2 </mn> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mrow> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> &#964;L </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 6 </mn> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 6 </mn> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mtext> </mtext> <msqrt> <mfrac> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 4 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 9 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 16 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 25 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </msqrt> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> &#964;Z </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> z </ci> </apply> </apply> <apply> <power /> <pi /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <ci> &#964;L </ci> <apply> <plus /> <cn type='integer'> 6 </cn> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 6 </cn> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <sinh /> <apply> <times /> <pi /> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <times /> <ci> z </ci> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 4 </cn> </apply> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 9 </cn> </apply> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 16 </cn> </apply> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 25 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["RamanujanTauZ", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]]], " ", SuperscriptBox["\[Pi]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", RowBox[List["\[ImaginaryI]", " ", "z"]]]]], " ", RowBox[List["RamanujanTauL", "[", RowBox[List["6", "+", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["6", "+", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "]"]], " ", SqrtBox[FractionBox[RowBox[List["Sinh", "[", RowBox[List["\[Pi]", " ", "z"]], "]"]], RowBox[List["z", " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["z", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List["4", "+", SuperscriptBox["z", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List["9", "+", SuperscriptBox["z", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List["16", "+", SuperscriptBox["z", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List["25", "+", SuperscriptBox["z", "2"]]], ")"]]]]]]]]]]]]










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





2007-05-02