Hurwitz zeta function: General characteristics (formula 10.02.04.0026)

Zeta

$Mathematica Notation$

Zeta Functions and Polylogarithms

Zeta[s,a]

General characteristics

Branch cuts

With respect to a

For zeta^{^}(s,a)

http://functions.wolfram.com/10.02.04.0026.01

Input Form

Limit[ZetaClassical[s, x + I \[Epsilon]], \[Epsilon] -> Plus[0]] == ZetaClassical[s, x] /; x < 0

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Limit", "[", RowBox[List[RowBox[List["ZetaClassical", "[", RowBox[List["s", ",", RowBox[List["x", "+", RowBox[List["\[ImaginaryI]", " ", "\[Epsilon]"]]]]]], "]"]], ",", RowBox[List["\[Epsilon]", "\[Rule]", RowBox[List["+", "0"]]]]]], "]"]], "\[Equal]", RowBox[List["ZetaClassical", "[", RowBox[List["s", ",", "x"]], "]"]]]], "/;", " ", RowBox[List["x", "<", "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> <mrow> <munder> <mi> lim </mi> <mrow> <mi> ϵ </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mrow> <mo> + </mo> <mn> 0 </mn> </mrow> </mrow> </munder> <mo> ⁢ </mo> <mtext>   </mtext> <semantics> <mrow> <mover> <mi> ζ </mi> <mo> ^ </mo> </mover> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> s </mi> <mo> , </mo> <mrow> <mi> x </mi> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> ϵ </mi> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[OverscriptBox["\[Zeta]", "^"], RowBox[List["(", RowBox[List[TagBox["s", Rule[Editable, True]], ",", TagBox[RowBox[List["x", "+", RowBox[List["\[ImaginaryI]", " ", "\[Epsilon]"]]]], Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[List[$CellContext`e1, $CellContext`e2], Zeta[$CellContext`e1, $CellContext`e2]]]] </annotation> </semantics> </mrow> <mo> ⩵ </mo> <semantics> <mrow> <mover> <mi> ζ </mi> <mo> ^ </mo> </mover> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> s </mi> <mo> , </mo> <mi> x </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[OverscriptBox["\[Zeta]", "^"], RowBox[List["(", RowBox[List[TagBox["s", Rule[Editable, True]], ",", TagBox["x", Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[List[$CellContext`e1, $CellContext`e2], Zeta[$CellContext`e1, $CellContext`e2]]]] </annotation> </semantics> </mrow> <mo> /; </mo> <mrow> <mi> x </mi> <mo> < </mo> <mn> 0 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <limit /> <bvar> <ci> ϵ </ci> </bvar> <condition> <apply> <tendsto /> <ci> ϵ </ci> <apply> <plus /> <cn type='integer'> 0 </cn> </apply> </apply> </condition> <apply> <ci> Zeta </ci> <ci> s </ci> <apply> <plus /> <ci> x </ci> <apply> <times /> <imaginaryi /> <ci> ϵ </ci> </apply> </apply> </apply> </apply> <apply> <ci> Zeta </ci> <ci> s </ci> <ci> x </ci> </apply> </apply> <apply> <lt /> <ci> x </ci> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Limit", "[", RowBox[List[RowBox[List["ZetaClassical", "[", RowBox[List["s_", ",", RowBox[List["x_", "+", RowBox[List["\[ImaginaryI]", " ", "\[Epsilon]_"]]]]]], "]"]], ",", RowBox[List["\[Epsilon]_", "\[Rule]", RowBox[List["+", "0"]]]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["ZetaClassical", "[", RowBox[List["s", ",", "x"]], "]"]], "/;", RowBox[List["x", "<", "0"]]]]]]]]

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

2001-10-29

Zeta[s]