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variants of this functions
Zeta




Mathematica Notation

Traditional Notation









Zeta Functions and Polylogarithms > Zeta[s,a] > Integral representations > Contour integral representations > Contour integral representations > For zeta^(s,a)





http://functions.wolfram.com/10.02.07.0003.01









  


  








Input Form





ZetaClassical[s, a] == (1/(2 Pi I)) ContourIntegrate[ (Gamma[t] Gamma[1 - t] Gamma[a - t]^s)/Gamma[1 + a - t]^s/(-1)^t, {t, \[ScriptCapitalL]}] /; Element[s, Integers] && s > 0








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ZetaClassical", "[", RowBox[List["s", ",", "a"]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["2", "\[Pi]", " ", "\[ImaginaryI]"]]], RowBox[List["ContourIntegrate", "[", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", "t", "]"]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "t"]], "]"]], SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List["a", "-", "t"]], "]"]], "s"]]], SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "-", "t"]], "]"]], "s"]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["-", "t"]]]]], ",", RowBox[List["{", RowBox[List["t", ",", "\[ScriptCapitalL]"]], "}"]]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["s", "\[Element]", "Integers"]], "\[And]", RowBox[List["s", ">", "0"]]]]]]]]








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> <mover> <mi> &#950; </mi> <mo> ^ </mo> </mover> <mo> ( </mo> <mrow> <mi> s </mi> <mo> , </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[OverscriptBox[&quot;\[Zeta]&quot;, &quot;^&quot;], &quot;(&quot;, RowBox[List[TagBox[&quot;s&quot;, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;a&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> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msub> <mo> &#8747; </mo> <mi> &#8466; </mi> </msub> <mrow> <mfrac> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> <mi> s </mi> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> t </mi> </mrow> </msup> </mrow> <msup> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> a </mi> <mo> - </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> <mi> s </mi> </msup> </mfrac> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> t </mi> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> s </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> FormBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> OverscriptBox </ci> <ms> &#950; </ms> <ms> ^ </ms> </apply> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <ms> s </ms> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <ms> a </ms> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> InterpretTemplate </ci> <lambda> <bvar> <ci> $CellContext`e1 </ci> </bvar> <bvar> <ci> $CellContext`e2 </ci> </bvar> <apply> <ci> Zeta </ci> <ci> $CellContext`e1 </ci> <ci> $CellContext`e2 </ci> </apply> </lambda> </apply> </apply> <ms> &#10869; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> &#960; </ms> <ms> &#8520; </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <apply> <ci> ErrorBox </ci> <ms> &#8747; </ms> </apply> <ms> &#8466; </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <ms> ( </ms> <ms> t </ms> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <ms> t </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> a </ms> <ms> - </ms> <ms> t </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> s </ms> </apply> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> t </ms> </list> </apply> </apply> </list> </apply> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> + </ms> <ms> a </ms> <ms> - </ms> <ms> t </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> s </ms> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#8518; </ms> <ms> t </ms> </list> </apply> </list> </apply> </list> </apply> </list> </apply> </list> </apply> <ms> /; </ms> <apply> <ci> RowBox </ci> <list> <ms> s </ms> <ms> &#8712; </ms> <apply> <ci> SuperscriptBox </ci> <ms> &#8469; </ms> <ms> + </ms> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ZetaClassical", "[", RowBox[List["s_", ",", "a_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["ContourIntegrate", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", "t", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "t"]], "]"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List["a", "-", "t"]], "]"]], "s"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["-", "t"]]]]], SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "-", "t"]], "]"]], "s"]], ",", RowBox[List["{", RowBox[List["t", ",", "\[ScriptCapitalL]"]], "}"]]]], "]"]], RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]"]]], "/;", RowBox[List[RowBox[List["s", "\[Element]", "Integers"]], "&&", RowBox[List["s", ">", "0"]]]]]]]]]]


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





2001-10-29