Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











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/10.02.06.0042.01









  


  








Input Form





ZetaClassical[s, a] \[Proportional] 1/(a + n)^s + Sum[1/(k - n + (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 + \[Ellipsis] /; (a -> -n) && Element[n, Integers] && n >= 0








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ZetaClassical", "[", RowBox[List["s", ",", "a"]], "]"]], "\[Proportional]", RowBox[List[FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "n"]], ")"]], "s"]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["k", "-", "n", "+", RowBox[List["(", RowBox[List["a", "+", "n"]], ")"]]]], ")"]], "s"]]]], "+", RowBox[List["Zeta", "[", "s", "]"]], "-", " ", RowBox[List["s", " ", RowBox[List["Zeta", "[", RowBox[List["1", "+", "s"]], "]"]], RowBox[List["(", RowBox[List["a", "+", "n"]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List["s", RowBox[List["(", RowBox[List["s", "+", "1"]], ")"]]]], "2"], RowBox[List["Zeta", "[", RowBox[List["2", "+", "s"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "n"]], ")"]], "2"]]], "+", "\[Ellipsis]"]]]], " ", "/;", RowBox[List[RowBox[List["(", RowBox[List["a", "\[Rule]", RowBox[List["-", "n"]]]], ")"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mover> <mi> &#950; </mi> <mo> ^ </mo> </mover> <mo> ( </mo> <mrow> <semantics> <mi> s </mi> <annotation encoding='Mathematica'> TagBox[&quot;s&quot;, Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <mi> a </mi> <annotation encoding='Mathematica'> TagBox[&quot;a&quot;, Rule[Editable, True]] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> <mo> &#8733; </mo> <mrow> <mfrac> <mn> 1 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mi> s </mi> </msup> </mfrac> <mo> + </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mfrac> <mn> 1 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> n </mi> <mo> + </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> s </mi> </msup> </mfrac> </mrow> <mo> + </mo> <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> - </mo> <mrow> <mi> s </mi> <mo> &#8290; </mo> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> s </mi> <mo> + </mo> <mn> 1 </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;1&quot;]], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mi> s </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> s </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mn> 2 </mn> </mfrac> <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;]], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </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> <mo> &#8230; </mo> </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> <apply> <ci> OverHat </ci> <ci> &#950; </ci> </apply> <apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> <ci> s </ci> </apply> <apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> <ci> a </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> a </ci> <ci> n </ci> </apply> <ci> s </ci> </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 /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <plus /> <ci> a </ci> <ci> n </ci> </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 /> <apply> <times /> <ci> s </ci> <apply> <plus /> <ci> s </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <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> <ci> &#8230; </ci> </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





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ZetaClassical", "[", RowBox[List["s_", ",", "a_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "n"]], ")"]], "s"]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["k", "-", "n", "+", RowBox[List["(", RowBox[List["a", "+", "n"]], ")"]]]], ")"]], "s"]]]], "+", RowBox[List["Zeta", "[", "s", "]"]], "-", RowBox[List["s", " ", RowBox[List["Zeta", "[", RowBox[List["1", "+", "s"]], "]"]], " ", RowBox[List["(", RowBox[List["a", "+", "n"]], ")"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["s", " ", RowBox[List["(", RowBox[List["s", "+", "1"]], ")"]]]], ")"]], " ", RowBox[List["Zeta", "[", RowBox[List["2", "+", "s"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "n"]], ")"]], "2"]]], "+", "\[Ellipsis]"]], "/;", RowBox[List[RowBox[List["(", RowBox[List["a", "\[Rule]", RowBox[List["-", "n"]]]], ")"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]


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





2007-05-02