Hurwitz zeta function: Integral representations (formula 10.02.07.0005)

Zeta

$Mathematica Notation$

Zeta Functions and Polylogarithms

Zeta[s,a]

Integral representations

Contour integral representations

For zeta^{^}(s,a)

http://functions.wolfram.com/10.02.07.0005.01

Input Form

ZetaClassical[s, a - n] == (Pi/(2 I (s - 1))) Integrate[(a - t)^(1 - s) Csc[Pi t]^2, {t, \[Gamma] - I Infinity, \[Gamma] + I Infinity}] /; Element[n, Integers] && n < \[Gamma] < n + 1 && Re[a] > \[Gamma] && s != 1

Standard Form

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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> ζ </mi> <mo> ^ </mo> </mover> <mo> ( </mo> <mrow> <mi> s </mi> <mo> , </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> n </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[OverscriptBox["\[Zeta]", "^"], "(", RowBox[List[TagBox["s", Rule[Editable, True]], ",", TagBox[RowBox[List["a", "-", "n"]], Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[List[$CellContext`e1, $CellContext`e2], Zeta[$CellContext`e1, $CellContext`e2]]]] </annotation> </semantics> <mo> ⩵ </mo> <mrow> <mfrac> <mi> π </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> s </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <msubsup> <mo> ∫ </mo> <mrow> <mi> γ </mi> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> ∞ </mi> </mrow> </mrow> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> ∞ </mi> </mrow> <mo> + </mo> <mi> γ </mi> </mrow> </msubsup> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> s </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <msup> <mi> csc </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> t </mi> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> n </mi> <mo> < </mo> <mi> γ </mi> <mo> < </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> > </mo> <mi> γ </mi> </mrow> <mo> ∧ </mo> <mrow> <mi> s </mi> <mo> ≠ </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> Zeta </ci> <ci> s </ci> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <pi /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <apply> <plus /> <ci> s </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <apply> <plus /> <ci> γ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <infinity /> </apply> </apply> </apply> </lowlimit> <uplimit> <apply> <plus /> <apply> <times /> <imaginaryi /> <infinity /> </apply> <ci> γ </ci> </apply> </uplimit> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> t </ci> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> </apply> <apply> <power /> <apply> <csc /> <apply> <times /> <pi /> <ci> t </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <integers /> </apply> <apply> <lt /> <ci> n </ci> <ci> γ </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <gt /> <apply> <real /> <ci> a </ci> </apply> <ci> γ </ci> </apply> <apply> <neq /> <ci> s </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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Contributed by

Allan Cortzen

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

2003-08-21

Zeta[s]