Hurwitz zeta function: Identities (formula 10.02.17.0014)

Zeta

$Mathematica Notation$

Zeta Functions and Polylogarithms

Zeta[s,a]

Identities

Functional identities

Major general cases

For zeta(s,a)

http://functions.wolfram.com/10.02.17.0014.01

Input Form

Zeta[s, a] == Zeta[s, Ceiling[-Re[a]] + a] + Zeta[s, -a - Floor[-Re[a]]] - Zeta[s, -a + 1] - KroneckerDelta[FractionalPart[-Re[a]]]/ (-Im[a]^2)^(s/2) /; !Element[a, Reals]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Zeta", "[", RowBox[List["s", ",", "a"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["Zeta", "[", RowBox[List["s", ",", RowBox[List[RowBox[List["Ceiling", "[", RowBox[List["-", RowBox[List["Re", "[", "a", "]"]]]], "]"]], "+", "a"]]]], "]"]], "+", RowBox[List["Zeta", "[", RowBox[List["s", ",", RowBox[List[RowBox[List["-", "a"]], "-", RowBox[List["Floor", "[", RowBox[List["-", RowBox[List["Re", "[", "a", "]"]]]], "]"]]]]]], "]"]], "-", RowBox[List["Zeta", "[", RowBox[List["s", ",", RowBox[List[RowBox[List["-", "a"]], "+", "1"]]]], "]"]], "-", RowBox[List[RowBox[List["KroneckerDelta", "[", RowBox[List["FractionalPart", "[", RowBox[List["-", RowBox[List["Re", "[", "a", "]"]]]], "]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox[RowBox[List["Im", "[", "a", "]"]], "2"]]], ")"]], RowBox[List["-", FractionBox["s", "2"]]]]]]]]]], "/;", RowBox[List["Not", "[", RowBox[List["a", "\[Element]", "Reals"]], "]"]]]]]]

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> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> s </mi> <mo> , </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", RowBox[List[TagBox["s", Zeta, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["a", Zeta, Rule[Editable, True], Rule[Selectable, True]]]], ")"]], InterpretTemplate[Function[List[ZetaDump`e1$, ZetaDump`e2$], Zeta[ZetaDump`e1$, ZetaDump`e2$]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo>  </mo> <mrow> <mrow> <mo> - </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> s </mi> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> a </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", RowBox[List[TagBox["s", Zeta, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["1", "-", "a"]], Zeta, Rule[Editable, True], Rule[Selectable, True]]]], ")"]], InterpretTemplate[Function[List[ZetaDump`e1$, ZetaDump`e2$], Zeta[ZetaDump`e1$, ZetaDump`e2$]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> </mrow> <mo> + </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> s </mi> <mo> , </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mo> ⌈ </mo> <mrow> <mo> - </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mrow> <mo> ⌉ </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", RowBox[List[TagBox["s", Zeta, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["a", "+", RowBox[List["\[LeftCeiling]", RowBox[List["-", RowBox[List["Re", "(", "a", ")"]]]], "\[RightCeiling]"]]]], Zeta, Rule[Editable, True], Rule[Selectable, True]]]], ")"]], InterpretTemplate[Function[List[ZetaDump`e1$, ZetaDump`e2$], Zeta[ZetaDump`e1$, ZetaDump`e2$]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> + </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> s </mi> <mo> , </mo> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> - </mo> <mrow> <mo> ⌊ </mo> <mrow> <mo> - </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mrow> <mo> ⌋ </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", RowBox[List[TagBox["s", Zeta, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[RowBox[List["-", "a"]], "-", RowBox[List["\[LeftFloor]", RowBox[List["-", RowBox[List["Re", "(", "a", ")"]]]], "\[RightFloor]"]]]], Zeta, Rule[Editable, True], Rule[Selectable, True]]]], ")"]], InterpretTemplate[Function[List[ZetaDump`e1$, ZetaDump`e2$], Zeta[ZetaDump`e1$, ZetaDump`e2$]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> - </mo> <mrow> <msub> <semantics> <mi> δ </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mi> frac </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </msub> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <msup> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mfrac> <mi> s </mi> <mn> 2 </mn> </mfrac> </mrow> </msup> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> a </mi> <mo> ∉ </mo> <semantics> <mi> ℝ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalR]", Function[List[], Reals]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> Zeta </ci> <ci> s </ci> <ci> a </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Zeta </ci> <ci> s </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> </apply> </apply> <apply> <ci> Zeta </ci> <ci> s </ci> <apply> <plus /> <ci> a </ci> <apply> <ceiling /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <real /> <ci> a </ci> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Zeta </ci> <ci> s </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <floor /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <real /> <ci> a </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> KroneckerDelta </ci> <apply> <ci> FractionalPart </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <real /> <ci> a </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <imaginary /> <ci> a </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> s </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <notin /> <ci> a </ci> <reals /> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Zeta", "[", RowBox[List["s_", ",", "a_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["Zeta", "[", RowBox[List["s", ",", RowBox[List[RowBox[List["Ceiling", "[", RowBox[List["-", RowBox[List["Re", "[", "a", "]"]]]], "]"]], "+", "a"]]]], "]"]], "+", RowBox[List["Zeta", "[", RowBox[List["s", ",", RowBox[List[RowBox[List["-", "a"]], "-", RowBox[List["Floor", "[", RowBox[List["-", RowBox[List["Re", "[", "a", "]"]]]], "]"]]]]]], "]"]], "-", RowBox[List["Zeta", "[", RowBox[List["s", ",", RowBox[List[RowBox[List["-", "a"]], "+", "1"]]]], "]"]], "-", RowBox[List[RowBox[List["KroneckerDelta", "[", RowBox[List["FractionalPart", "[", RowBox[List["-", RowBox[List["Re", "[", "a", "]"]]]], "]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox[RowBox[List["Im", "[", "a", "]"]], "2"]]], ")"]], RowBox[List["-", FractionBox["s", "2"]]]]]]]], "/;", RowBox[List["!", RowBox[List["a", "\[Element]", "Reals"]]]]]]]]]]

Contributed by

Pavlyk 0. (2006)

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

2007-05-02

Zeta[s]