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variants of this functions
Zeta






Mathematica Notation

Traditional Notation









Zeta Functions and Polylogarithms > Zeta[s] > Series representations > Generalized power series > Expansions at s==1





http://functions.wolfram.com/10.01.06.0026.01









  


  










Input Form





Derivative[1][Zeta][s]/Zeta[s] == -(1/(s - 1)) - Sum[Subscript[\[Eta], k] (s - 1)^k, {k, 0, Infinity}] /; (Subscript[\[Eta], k] = (k + 1) Sum[((-1)^(j + 1)/(j + 1)) Subscript[c, k - j, j + 1], {j, 0, k}] /; (Subscript[c, 0, k] = EulerGamma^k && Subscript[c, m, k] = (1/(m EulerGamma)) Sum[(((k m - (k + 1) i) (-1)^(m - i))/(m - i)!) StieltjesGamma[m - i] Subscript[c, i, k], {i, 0, m - 1}]))










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["Zeta", "\[Prime]", Rule[MultilineFunction, None]], "[", "s", "]"]], RowBox[List["Zeta", "[", "s", "]"]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", RowBox[List["s", "-", "1"]]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[SubscriptBox["\[Eta]", "k"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["s", "-", "1"]], ")"]], "k"]]]]]]]]], "/;", RowBox[List["(", RowBox[List[SubscriptBox["\[Eta]", "k"], "=", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["j", "+", "1"]]], " "]], RowBox[List["j", "+", "1"]]], SubscriptBox["c", RowBox[List[RowBox[List["k", "-", "j"]], ",", RowBox[List["j", "+", "1"]]]]]]]]]]], "/;", RowBox[List["(", RowBox[List[SubscriptBox["c", RowBox[List["0", ",", "k"]]], "=", RowBox[List[RowBox[List[SuperscriptBox["EulerGamma", "k"], "\[And]", SubscriptBox["c", RowBox[List["m", ",", "k"]]]]], "=", RowBox[List[FractionBox["1", RowBox[List["m", " ", "EulerGamma"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], RowBox[List["m", "-", "1"]]], RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["k", " ", "m"]], "-", RowBox[List[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], " ", "i"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["m", "-", "i"]]]]], RowBox[List[RowBox[List["(", RowBox[List["m", "-", "i"]], ")"]], "!"]]], " ", RowBox[List["StieltjesGamma", "[", RowBox[List["m", "-", "i"]], "]"]], " ", SubscriptBox["c", RowBox[List["i", ",", "k"]]]]]]]]]]]]], ")"]]]]]], ")"]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mfrac> <mrow> <msup> <mi> &#950; </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, TagBox[&quot;s&quot;, Zeta, Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> </mfrac> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mrow> <mi> s </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> <mo> - </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <msub> <mi> &#951; </mi> <mi> k </mi> </msub> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> s </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> &#951; </mi> <mi> k </mi> </msub> <mo> = </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> k </mi> </munderover> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> j </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mtext> </mtext> </mrow> <mrow> <mi> j </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> &#8290; </mo> <msub> <mi> c </mi> <mrow> <mrow> <mi> k </mi> <mo> - </mo> <mi> j </mi> </mrow> <mo> , </mo> <mrow> <mi> j </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> </msub> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> c </mi> <mrow> <mn> 0 </mn> <mo> , </mo> <mi> k </mi> </mrow> </msub> <mo> = </mo> <mrow> <mrow> <msup> <semantics> <mi> &#8509; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubledGamma]&quot;, Function[List[], EulerGamma]] </annotation> </semantics> <mi> k </mi> </msup> <mo> &#8743; </mo> <msub> <mi> c </mi> <mrow> <mi> m </mi> <mo> , </mo> <mi> k </mi> </mrow> </msub> </mrow> <mo> = </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mi> m </mi> <mo> &#8290; </mo> <semantics> <mi> &#8509; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubledGamma]&quot;, Function[List[], EulerGamma]] </annotation> </semantics> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> i </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> k </mi> <mo> &#8290; </mo> <mi> m </mi> </mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> i </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> m </mi> <mo> - </mo> <mi> i </mi> </mrow> </msup> <mtext> </mtext> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mi> i </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <msub> <semantics> <mi> &#947; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Gamma]&quot;, StieltjesGamma] </annotation> </semantics> <mrow> <mi> m </mi> <mo> - </mo> <mi> i </mi> </mrow> </msub> <mo> &#8290; </mo> <msub> <mi> c </mi> <mrow> <mi> i </mi> <mo> , </mo> <mi> k </mi> </mrow> </msub> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <times /> <apply> <partialdiff /> <bvar> <ci> s </ci> </bvar> <apply> <ci> Zeta </ci> <ci> s </ci> </apply> </apply> <apply> <power /> <apply> <ci> Zeta </ci> <ci> s </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <ci> s </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <ci> Subscript </ci> <ci> &#951; </ci> <ci> k </ci> </apply> <apply> <power /> <apply> <plus /> <ci> s </ci> <cn type='integer'> -1 </cn> </apply> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Set </ci> <apply> <ci> Subscript </ci> <ci> &#951; </ci> <ci> k </ci> </apply> <apply> <ci> Condition </ci> <apply> <times /> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> k </ci> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> c </ci> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> <apply> <plus /> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Set </ci> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 0 </cn> <ci> k </ci> </apply> <apply> <ci> Set </ci> <apply> <and /> <apply> <power /> <eulergamma /> <ci> k </ci> </apply> <apply> <ci> Subscript </ci> <ci> c </ci> <ci> m </ci> <ci> k </ci> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <ci> m </ci> <eulergamma /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> i </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> k </ci> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <ci> i </ci> </apply> </apply> </apply> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> i </ci> </apply> </apply> </apply> <apply> <power /> <apply> <factorial /> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> i </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> StieltjesGamma </ci> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> i </ci> </apply> </apply> </apply> <apply> <ci> Subscript </ci> <ci> c </ci> <ci> i </ci> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", FractionBox[RowBox[List[SuperscriptBox["Zeta", "\[Prime]", Rule[MultilineFunction, None]], "[", "s_", "]"]], RowBox[List["Zeta", "[", "s_", "]"]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", RowBox[List["s", "-", "1"]]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[SubscriptBox["\[Eta]", "k"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["s", "-", "1"]], ")"]], "k"]]]]]]], "/;", RowBox[List["(", RowBox[List[SubscriptBox["\[Eta]", "k"], "=", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["j", "+", "1"]]], " ", SubscriptBox["c", RowBox[List[RowBox[List["k", "-", "j"]], ",", RowBox[List["j", "+", "1"]]]]]]], RowBox[List["j", "+", "1"]]]]]]], "/;", RowBox[List["(", RowBox[List[SubscriptBox["c", RowBox[List["0", ",", "k"]]], "=", RowBox[List[RowBox[List[SuperscriptBox["EulerGamma", "k"], "&&", SubscriptBox["c", RowBox[List["m", ",", "k"]]]]], "=", FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], RowBox[List["m", "-", "1"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["k", " ", "m"]], "-", RowBox[List[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], " ", "i"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["m", "-", "i"]]]]], ")"]], " ", RowBox[List["StieltjesGamma", "[", RowBox[List["m", "-", "i"]], "]"]], " ", SubscriptBox["c", RowBox[List["i", ",", "k"]]]]], RowBox[List[RowBox[List["(", RowBox[List["m", "-", "i"]], ")"]], "!"]]]]], RowBox[List["m", " ", "EulerGamma"]]]]]]], ")"]]]]]], ")"]]]]]]]]










Contributed by





Krzysztof Maslanka










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





2007-05-02





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