Zeta[s] \[Proportional] 1/(s - 1) + EulerGamma - StieltjesGamma[1] (s - 1) + (1/2) StieltjesGamma[2] (s - 1)^2 + O[(s - 1)^3]

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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> <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;, Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> <mo> &#8733; </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mi> s </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> + </mo> <semantics> <mi> &#8509; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubledGamma]&quot;, Function[EulerGamma]] </annotation> </semantics> <mo> - </mo> <mrow> <msub> <semantics> <mi> &#947; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Gamma]&quot;, StieltjesGamma] </annotation> </semantics> <mn> 1 </mn> </msub> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> s </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <msub> <semantics> <mi> &#947; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Gamma]&quot;, StieltjesGamma] </annotation> </semantics> <mn> 2 </mn> </msub> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> s </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> O </mi> <mo> &#8289; </mo> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> s </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Proportional </ci> <apply> <ci> Zeta </ci> <ci> s </ci> </apply> <apply> <plus /> <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> <eulergamma /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> StieltjesGamma </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> s </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <ci> StieltjesGamma </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> s </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> O </ci> <apply> <power /> <apply> <plus /> <ci> s </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2007-05-02