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variants of this functions
Zeta




Mathematica Notation

Traditional Notation









Zeta Functions and Polylogarithms > Zeta[s,a] > Series representations > Other series representations > Other series representations > For zeta^(s,a)





http://functions.wolfram.com/10.02.06.0020.01









  


  








Input Form





ZetaClassical[s, a] == Sum[(k + a)^(-s), {k, 0, n}] + (a + n)^(1 - s)/(s - 1) + Sum[(Pochhammer[s + k - 1, k] BernoulliB[k + 1])/ ((k + 1)! (a + n)^(s + k)), {k, 0, r - 1}] - (Pochhammer[s, r + 1]/(r + 1)!) Integrate[ (BernoulliB[r + 1, t - Floor[t]] - BernoulliB[r + 1])/ (t + a + n)^(s + r + 1), {t, 0, Infinity}] /; Element[n, Integers] && n >= 0 && Element[r, Integers] && r >= 0 && Re[a] > -n && !(Element[-a, Integers] && -a > 0) && Re[s] > -r && s != 1








Standard Form





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MathML Form







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</mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> r </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mfrac> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> s </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;k&quot;, &quot;+&quot;, &quot;s&quot;, &quot;-&quot;, &quot;1&quot;]], &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <msub> <semantics> <mi> B </mi> <annotation encoding='Mathematica'> TagBox[&quot;B&quot;, BernoulliB] </annotation> </semantics> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> </mrow> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> + </mo> <mi> s </mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> <mrow> <mi> r </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, &quot;s&quot;, &quot;)&quot;]], RowBox[List[&quot;r&quot;, &quot;+&quot;, &quot;1&quot;]]], Pochhammer] </annotation> </semantics> <mtext> </mtext> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> r </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; 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Rule Form





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Contributed by





Allan Cortzen


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





2002-12-18