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variants of this functions
Zeta






Mathematica Notation

Traditional Notation









Zeta Functions and Polylogarithms > Zeta[s,a] > Series representations > Generalized power series > Expansions at a==-n > For zeta(s,a)





http://functions.wolfram.com/10.02.06.0039.01









  


  










Input Form





Zeta[s, a] \[Proportional] 1/((a + n)^2)^(s/2) + Sum[1/(n - k - (a + n))^s, {k, 0, n - 1}] + Zeta[s] - s Zeta[1 + s] (a + n) + ((s (s + 1))/2) Zeta[2 + s] (a + n)^2 + O[(a + n)^3] /; (a -> -n) && Element[n, Integers] && n >= 0










Standard Form





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MathML Form







<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> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> s </mi> <mo> , </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, RowBox[List[TagBox[&quot;s&quot;, Zeta, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[&quot;a&quot;, Zeta, Rule[Editable, True], Rule[Selectable, True]]]], &quot;)&quot;]], InterpretTemplate[Function[List[ZetaDump`e1$, ZetaDump`e2$], Zeta[ZetaDump`e1$, ZetaDump`e2$]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> &#8733; </mo> <mrow> <mfrac> <mn> 1 </mn> <msup> <mrow> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ) </mo> </mrow> <mrow> <mi> s </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mfrac> <mo> + </mo> <mrow> <munderover> <mo> &#8721; 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</mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> s </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> s </mi> <mo> &#8290; </mo> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> s </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, TagBox[RowBox[List[&quot;s&quot;, &quot;+&quot;, &quot;2&quot;]], Zeta, Rule[Editable, True], Rule[Selectable, True]], &quot;)&quot;]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> n </mi> </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> a </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mrow> <mo> - </mo> <mi> n </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <mi> &#8469; </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 /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> a </ci> <ci> n </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> <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> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> a </ci> <ci> n </ci> </apply> </apply> </apply> <ci> s </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Zeta </ci> <ci> s </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> s </ci> <apply> <ci> Zeta </ci> <apply> <plus /> <ci> s </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <plus /> <ci> a </ci> <ci> n </ci> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> s </ci> <cn type='integer'> 1 </cn> </apply> <ci> s </ci> <apply> <ci> Zeta </ci> <apply> <plus /> <ci> s </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <ci> n </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> O </ci> <apply> <power /> <apply> <plus /> <ci> a </ci> <ci> n </ci> </apply> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <ci> Rule </ci> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <apply> <in /> <ci> n </ci> <ci> &#8469; </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2007-05-02