|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
http://functions.wolfram.com/10.02.04.0018.01
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
BranchCuts[Zeta[s, a], a] ==
{SequenceList[{Interval[{(-I) Infinity - n, -n}], 1},
Element[n, Integers] && n >= 0], SequenceList[
{Interval[{-n, -n + I Infinity}], -1}, Element[n, Integers] && n >= 0]}
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Cell[BoxData[RowBox[List[RowBox[List["BranchCuts", "[", RowBox[List[RowBox[List["Zeta", "[", RowBox[List["s", ",", "a"]], "]"]], ",", "a"]], "]"]], "\[Equal]", RowBox[List["{", RowBox[List[RowBox[List["SequenceList", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[Infinity]"]], "-", "n"]], ",", RowBox[List["-", "n"]]]], "}"]], "]"]], ",", "1"]], "}"]], ",", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]], "]"]], ",", RowBox[List["SequenceList", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "n"]], ",", RowBox[List[RowBox[List["-", "n"]], "+", RowBox[List["\[ImaginaryI]", " ", "\[Infinity]"]]]]]], "}"]], "]"]], ",", RowBox[List["-", "1"]]]], "}"]], ",", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]], "]"]]]], "}"]]]]]]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msub> <mi> ℬ𝒞 </mi> <mi> a </mi> </msub> <mo> ( </mo> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <semantics> <mi> s </mi> <annotation encoding='Mathematica'> TagBox["s", Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <mi> a </mi> <annotation encoding='Mathematica'> TagBox["a", Rule[Editable, True]] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mo> { </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> ∞ </mi> </mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mi> n </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> } </mo> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> </mrow> <mo> } </mo> </mrow> <mo> , </mo> <mrow> <mo> { </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> , </mo> <mrow> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> ∞ </mi> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> } </mo> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> </mrow> <mo> } </mo> </mrow> </mrow> <mo> } </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <msub> <mi> ℬ𝒞 </mi> <mi> a </mi> </msub> <mo> ( </mo> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <semantics> <mi> s </mi> <annotation encoding='Mathematica'> TagBox["s", Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <mi> a </mi> <annotation encoding='Mathematica'> TagBox["a", Rule[Editable, True]] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mo> { </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> ∞ </mi> </mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mi> n </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> } </mo> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> </mrow> <mo> } </mo> </mrow> <mo> , </mo> <mrow> <mo> { </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> , </mo> <mrow> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> ∞ </mi> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> } </mo> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> </mrow> <mo> } </mo> </mrow> </mrow> <mo> } </mo> </mrow> </mrow> </annotation-xml> </semantics> </math>
|
|
|
|
|
|
|
|
|
|
| |
|
|
|
|
| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["BranchCuts", "[", RowBox[List[RowBox[List["Zeta", "[", RowBox[List["s_", ",", "a_"]], "]"]], ",", "a_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["{", RowBox[List[RowBox[List["SequenceList", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[Infinity]"]], "-", "n"]], ",", RowBox[List["-", "n"]]]], "}"]], "]"]], ",", "1"]], "}"]], ",", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]], "]"]], ",", RowBox[List["SequenceList", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "n"]], ",", RowBox[List[RowBox[List["-", "n"]], "+", RowBox[List["\[ImaginaryI]", " ", "\[Infinity]"]]]]]], "}"]], "]"]], ",", RowBox[List["-", "1"]]]], "}"]], ",", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]], "]"]]]], "}"]]]]]] |
|
|
|
|
|
|
|
|
|
|
Date Added to functions.wolfram.com (modification date)
|
|
|
|
|
|
|
|
|
|
|
|
|
|