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variants of this functions
Zeta






Mathematica Notation

Traditional Notation









Zeta Functions and Polylogarithms > Zeta[s,a] > Identities > Functional identities > Major general cases > For zeta(s,a)





http://functions.wolfram.com/10.02.17.0009.01









  


  










Input Form





Zeta[s, n/m] == ((2 Gamma[1 - s])/(2 Pi m)^(1 - s)) Sum[Sin[(Pi s)/2 + (2 Pi j n)/m] Zeta[1 - s, j/m], {j, 1, m}] /; Element[n, Integers] && n > 0 && Element[m, Integers] && m > 0 && n <= m










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Zeta", "[", RowBox[List["s", ",", FractionBox["n", "m"]]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["2", " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "s"]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "\[Pi]", " ", "m"]], ")"]], RowBox[List["1", "-", "s"]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "m"], RowBox[List[RowBox[List["Sin", "[", RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", "s"]], "2"], "+", FractionBox[RowBox[List["2", " ", "\[Pi]", " ", "j", " ", "n"]], "m"]]], "]"]], " ", RowBox[List["Zeta", "[", RowBox[List[RowBox[List["1", "-", "s"]], ",", FractionBox["j", "m"]]], "]"]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]], "\[And]", RowBox[List["n", "\[LessEqual]", "m"]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> s </mi> <mo> , </mo> <mfrac> <mi> n </mi> <mi> m </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[&quot;s&quot;, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;n&quot;, &quot;m&quot;], Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[List[$CellContext`e1, $CellContext`e2], Zeta[$CellContext`e1, $CellContext`e2]]]] </annotation> </semantics> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#915; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> s </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> s </mi> </mrow> </msup> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> m </mi> </munderover> <mrow> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> j </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mi> m </mi> </mfrac> <mo> + </mo> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> s </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> s </mi> </mrow> <mo> , </mo> <mfrac> <mi> j </mi> <mi> m </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, &quot;s&quot;]], Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;j&quot;, &quot;m&quot;], Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[List[$CellContext`e1, $CellContext`e2], Zeta[$CellContext`e1, $CellContext`e2]]]] </annotation> </semantics> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &#8804; </mo> <mi> m </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> Zeta </ci> <ci> s </ci> <apply> <times /> <ci> n </ci> <apply> <power /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#915; </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <ci> m </ci> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> m </ci> </uplimit> <apply> <times /> <apply> <sin /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> j </ci> <pi /> <ci> n </ci> <apply> <power /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <pi /> <ci> s </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <ci> Zeta </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> <apply> <times /> <ci> j </ci> <apply> <power /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> <apply> <in /> <ci> m </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> <apply> <leq /> <ci> n </ci> <ci> m </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Zeta", "[", RowBox[List["s_", ",", FractionBox["n_", "m_"]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "s"]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "m"], RowBox[List[RowBox[List["Sin", "[", RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", "s"]], "2"], "+", FractionBox[RowBox[List["2", " ", "\[Pi]", " ", "j", " ", "n"]], "m"]]], "]"]], " ", RowBox[List["Zeta", "[", RowBox[List[RowBox[List["1", "-", "s"]], ",", FractionBox["j", "m"]]], "]"]]]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "\[Pi]", " ", "m"]], ")"]], RowBox[List["1", "-", "s"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]], "&&", RowBox[List["n", "\[LessEqual]", "m"]]]]]]]]]]










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





2001-10-29





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