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http://functions.wolfram.com/10.02.20.0044.01
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Derivative[1, 0][Zeta][s, a] ==
(-(1/2)) Sum[Log[(a + k)^2]/((a + k)^2)^(s/2), {k, 0, Re[b - a] - 1}] +
Derivative[1, 0][Zeta][s, b - Min[0, Re[b - a]]] /;
Element[Re[b - a], Integers] && Im[b - a] == 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["Zeta", TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List["s", ",", "a"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["Re", "[", RowBox[List["b", "-", "a"]], "]"]], "-", "1"]]], FractionBox[RowBox[List["Log", "[", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "k"]], ")"]], "2"], "]"]], SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "k"]], ")"]], "2"], ")"]], RowBox[List["s", "/", "2"]]]]]]]], "+", RowBox[List[SuperscriptBox["Zeta", TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List["s", ",", RowBox[List["b", "-", RowBox[List["Min", "[", RowBox[List["0", ",", RowBox[List["Re", "[", RowBox[List["b", "-", "a"]], "]"]]]], "]"]]]]]], "]"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", RowBox[List["b", "-", "a"]], "]"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["Im", "[", RowBox[List["b", "-", "a"]], "]"]], "\[Equal]", "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> <mrow> <msup> <mi> ζ </mi> <semantics> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mrow> <mi> s </mi> <mo> , </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mo>  </mo> <mrow> <mrow> <msup> <mi> ζ </mi> <semantics> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mrow> <mi> s </mi> <mo> , </mo> <mrow> <mi> b </mi> <mo> - </mo> <mrow> <mi> min </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mfrac> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ) </mo> </mrow> <msup> <mrow> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ) </mo> </mrow> <mrow> <mi> s </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mfrac> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mo>  </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <apply> <partialdiff /> <list> <cn type='integer'> 1 </cn> <cn type='integer'> 0 </cn> </list> <ci> Zeta </ci> </apply> <ci> s </ci> <ci> a </ci> </apply> <apply> <plus /> <apply> <apply> <partialdiff /> <list> <cn type='integer'> 1 </cn> <cn type='integer'> 0 </cn> </list> <ci> Zeta </ci> </apply> <ci> s </ci> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <min /> <cn type='integer'> 0 </cn> <apply> <real /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <apply> <real /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <apply> <ln /> <apply> <power /> <apply> <plus /> <ci> a </ci> <ci> k </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> a </ci> <ci> k </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <ci> s </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <apply> <real /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> </apply> <integers /> </apply> <apply> <eq /> <apply> <imaginary /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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Date Added to functions.wolfram.com (modification date)
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