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variants of this functions
Zeta




Mathematica Notation

Traditional Notation









Zeta Functions and Polylogarithms > Zeta[s] > Specific values > Values at fixed points





http://functions.wolfram.com/10.01.03.0024.01









  


  








Input Form





Zeta[9] == (125 Pi^9)/3704778 - (992/495) Sum[1/(k^9 (E^(2 Pi k) - 1)), {k, 1, Infinity}] - (2/495) Sum[1/(k^9 (E^(2 Pi k) + 1)), {k, 1, Infinity}]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["Zeta", "[", "9", "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["125", SuperscriptBox["\[Pi]", "9"]]], "3704778"], "-", RowBox[List[FractionBox["992", "495"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox["1", RowBox[List[SuperscriptBox["k", "9"], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", "\[Pi]", " ", "k"]]], "-", "1"]], ")"]]]]]]]]], "-", RowBox[List[FractionBox["2", "495"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox["1", RowBox[List[SuperscriptBox["k", "9"], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", "\[Pi]", " ", "k"]]], "+", "1"]], ")"]]]]]]]]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 9 </mn> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, TagBox[&quot;9&quot;, Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[$CellContext`e, Zeta[$CellContext`e]]]] </annotation> </semantics> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <mn> 125 </mn> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 9 </mn> </msup> </mrow> <mn> 3704778 </mn> </mfrac> <mo> - </mo> <mrow> <mfrac> <mn> 992 </mn> <mn> 495 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mfrac> <mn> 1 </mn> <mrow> <msup> <mi> k </mi> <mn> 9 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> k </mi> </mrow> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 2 </mn> <mn> 495 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mfrac> <mn> 1 </mn> <mrow> <msup> <mi> k </mi> <mn> 9 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> k </mi> </mrow> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> Zeta </ci> <cn type='integer'> 9 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 125 </cn> <apply> <power /> <pi /> <cn type='integer'> 9 </cn> </apply> <apply> <power /> <cn type='integer'> 3704778 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 992 <sep /> 495 </cn> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <apply> <power /> <ci> k </ci> <cn type='integer'> 9 </cn> </apply> <apply> <plus /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <ci> k </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 2 <sep /> 495 </cn> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <apply> <power /> <ci> k </ci> <cn type='integer'> 9 </cn> </apply> <apply> <plus /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <ci> k </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Zeta", "[", "9", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["125", " ", SuperscriptBox["\[Pi]", "9"]]], "3704778"], "-", RowBox[List[FractionBox["992", "495"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox["1", RowBox[List[SuperscriptBox["k", "9"], " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[Pi]", " ", "k"]]], "-", "1"]], ")"]]]]]]]]], "-", RowBox[List[FractionBox["2", "495"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox["1", RowBox[List[SuperscriptBox["k", "9"], " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[Pi]", " ", "k"]]], "+", "1"]], ")"]]]]]]]]]]]]]]]


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





2001-10-29