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http://functions.wolfram.com/10.02.06.0046.01
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Zeta[s, a] \[Proportional] (1 + O[a - Subscript[a, 0]])/
(I^s E^(I Pi s Floor[(-(1/Pi)) (Arg[(a + n)/(Subscript[a, 0] + n)] +
Arg[(-I) (Subscript[a, 0] + n)])]) ((-I) (Subscript[a, 0] + n))^
s) + (Sum[1/(-Subscript[a, 0] - k)^s, {k, 0, n - 1}] +
Zeta[s, 1 + Subscript[a, 0] + n]) (1 + O[a - Subscript[a, 0]]) /;
Re[Subscript[a, 0]] == -n && Element[n, Integers] && n >= 0
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<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]], ",", TagBox["a", Zeta, Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[List[ZetaDump`e1$, ZetaDump`e2$], Zeta[ZetaDump`e1$, ZetaDump`e2$]]]] </annotation> </semantics> <mo> ∝ </mo> <mrow> <mrow> <msup> <mi> ⅈ </mi> <mrow> <mo> - </mo> <mi> s </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> s </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> s </mi> <mo> ⁢ </mo> <mrow> <mo> ⌊ </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mi> π </mi> </mfrac> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mi> n </mi> </mrow> <mrow> <mi> n </mi> <mo> + </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⌋ </mo> </mrow> </mrow> </msup> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> O </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mtext> </mtext> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mfrac> <mn> 1 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> k </mi> </mrow> <mo> - </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mi> s </mi> </msup> </mfrac> </mrow> <mo> + </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> s </mi> <mo> , </mo> <mrow> <mi> n </mi> <mo> + </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", RowBox[List[TagBox["s", Zeta, Rule[Editable, True]], ",", TagBox[RowBox[List["n", "+", SubscriptBox["a", "0"], "+", "1"]], Zeta, Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[List[ZetaDump`e1$, ZetaDump`e2$], Zeta[ZetaDump`e1$, ZetaDump`e2$]]]] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> O </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <mo>  </mo> <mrow> <mo> - </mo> <mi> n </mi> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> Zeta </ci> <ci> s </ci> <ci> a </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <imaginaryi /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <apply> <plus /> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> <apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <pi /> <ci> s </ci> <apply> <floor /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <arg /> <apply> <times /> <apply> <plus /> <ci> a </ci> <ci> n </ci> </apply> <apply> <power /> <apply> <plus /> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <arg /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <apply> <plus /> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <ci> O </ci> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <ci> s </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Zeta </ci> <ci> s </ci> <apply> <plus /> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <ci> O </ci> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <real /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <apply> <in /> <ci> n </ci> <ci> ℕ </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Zeta", "[", RowBox[List["s_", ",", "a_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["\[ImaginaryI]", RowBox[List["-", "s"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[Pi]", " ", "s", " ", RowBox[List["Floor", "[", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["Arg", "[", FractionBox[RowBox[List["a", "+", "n"]], RowBox[List[SubscriptBox["aa", "0"], "+", "n"]]], "]"]], "+", RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List[SubscriptBox["aa", "0"], "+", "n"]], ")"]]]], "]"]]]], "\[Pi]"]]], "]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List[SubscriptBox["aa", "0"], "+", "n"]], ")"]]]], ")"]], RowBox[List["-", "s"]]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", RowBox[List["a", "-", SubscriptBox["aa", "0"]]], "]"]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", SubscriptBox["aa", "0"]]], "-", "k"]], ")"]], "s"]]]], "+", RowBox[List["Zeta", "[", RowBox[List["s", ",", RowBox[List["1", "+", SubscriptBox["aa", "0"], "+", "n"]]]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", RowBox[List["a", "-", SubscriptBox["aa", "0"]]], "]"]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", SubscriptBox["aa", "0"], "]"]], "\[Equal]", RowBox[List["-", "n"]]]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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