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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/10.01.03.0058.01









  


  








Input Form





Zeta[5] == (Pi^4/45) (Log[2 Pi] - 120 (12 PolyGamma[-6, 1] - 6 PolyGamma[-5, 1] + PolyGamma[-4, 1]))








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["Zeta", "[", "5", "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[Pi]", "4"], " "]], "45"], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]], "-", RowBox[List["120", " ", RowBox[List["(", RowBox[List[RowBox[List["12", " ", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "6"]], ",", "1"]], "]"]]]], "-", RowBox[List["6", " ", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "5"]], ",", "1"]], "]"]]]], "+", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "4"]], ",", "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> 5 </mn> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, TagBox[&quot;5&quot;, Zeta, Rule[Editable, True], Rule[Selectable, True]], &quot;)&quot;]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> &#63449; </mo> <mrow> <mfrac> <mrow> <msup> <mi> &#960; </mi> <mn> 4 </mn> </msup> <mtext> </mtext> </mrow> <mn> 45 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mn> 120 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 12 </mn> <mo> &#8290; </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 6 </mn> </mrow> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 5 </mn> </mrow> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 4 </mn> </mrow> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> Zeta </ci> <cn type='integer'> 5 </cn> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <pi /> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <cn type='integer'> 45 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <ln /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 120 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 12 </cn> <apply> <ci> PolyGamma </ci> <cn type='integer'> -6 </cn> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <ci> PolyGamma </ci> <cn type='integer'> -5 </cn> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <ci> PolyGamma </ci> <cn type='integer'> -4 </cn> <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", "[", "5", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "45"], " ", SuperscriptBox["\[Pi]", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]], "-", RowBox[List["120", " ", RowBox[List["(", RowBox[List[RowBox[List["12", " ", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "6"]], ",", "1"]], "]"]]]], "-", RowBox[List["6", " ", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "5"]], ",", "1"]], "]"]]]], "+", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "4"]], ",", "1"]], "]"]]]], ")"]]]]]], ")"]]]]]]]]


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





2007-05-02