Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

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 s==1 > For zeta^(s,a)





http://functions.wolfram.com/10.02.06.0030.01









  


  










Input Form





ZetaClassical[s, a] \[Proportional] 1/(s - 1) - PolyGamma[a] + (-StieltjesGamma[1] - Sum[Log[k + a - 1]^j/(k + a - 1) - Log[k]^j/k, {k, 1, Infinity}]) (s - 1) + (1/2) (StieltjesGamma[2] - Sum[Log[k + a - 1]^j/(k + a - 1) - Log[k]^j/k, {k, 1, Infinity}]) (s - 1)^2 + O[(s - 1)^3]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["ZetaClassical", "[", RowBox[List["s", ",", "a"]], "]"]], "\[Proportional]", RowBox[List[FractionBox["1", RowBox[List["s", "-", "1"]]], "-", RowBox[List["PolyGamma", "[", "a", "]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["StieltjesGamma", "[", "1", "]"]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List["(", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["Log", "[", RowBox[List["k", "+", "a", "-", "1"]], "]"]], "j"], RowBox[List["k", "+", "a", "-", "1"]]], "-", FractionBox[SuperscriptBox[RowBox[List["Log", "[", "k", "]"]], "j"], "k"]]], ")"]]]]]], ")"]], RowBox[List["(", RowBox[List["s", "-", "1"]], ")"]]]], "+", RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List[RowBox[List["StieltjesGamma", "[", "2", "]"]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List["(", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["Log", "[", RowBox[List["k", "+", "a", "-", "1"]], "]"]], "j"], RowBox[List["k", "+", "a", "-", "1"]]], "-", FractionBox[SuperscriptBox[RowBox[List["Log", "[", "k", "]"]], "j"], "k"]]], ")"]]]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["s", "-", "1"]], ")"]], "2"]]], " ", "+", RowBox[List["O", "[", SuperscriptBox[RowBox[List["(", RowBox[List["s", "-", "1"]], ")"]], "3"], "]"]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <semantics> <mrow> <mover> <mi> &#950; </mi> <mo> ^ </mo> </mover> <mo> ( </mo> <mrow> <mi> s </mi> <mo> , </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[OverscriptBox[&quot;\[Zeta]&quot;, &quot;^&quot;], &quot;(&quot;, RowBox[List[TagBox[&quot;s&quot;, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;a&quot;, Rule[Editable, True]]]], &quot;)&quot;]], InterpretTemplate[Function[List[$CellContext`e1, $CellContext`e2], Zeta[$CellContext`e1, $CellContext`e2]]]] </annotation> </semantics> <mo> &#8733; </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mi> s </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> - </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <msub> <semantics> <mi> &#947; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Gamma]&quot;, StieltjesGamma] </annotation> </semantics> <mn> 1 </mn> </msub> </mrow> <mo> - </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <msup> <mi> log </mi> <mi> j </mi> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> a </mi> <mo> + </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> - </mo> <mfrac> <mrow> <msup> <mi> log </mi> <mi> j </mi> </msup> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> <mi> k </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <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> <mrow> <mo> ( </mo> <mrow> <msub> <semantics> <mi> &#947; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Gamma]&quot;, StieltjesGamma] </annotation> </semantics> <mn> 2 </mn> </msub> <mo> - </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <msup> <mi> log </mi> <mi> j </mi> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> a </mi> <mo> + </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> - </mo> <mfrac> <mrow> <msup> <mi> log </mi> <mi> j </mi> </msup> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> <mi> k </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <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> <ci> a </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> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <ci> a </ci> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> StieltjesGamma </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <ln /> <apply> <plus /> <ci> a </ci> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> j </ci> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <ln /> <ci> k </ci> </apply> <ci> j </ci> </apply> <apply> <power /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <ci> s </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <ci> StieltjesGamma </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <ln /> <apply> <plus /> <ci> a </ci> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> j </ci> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <ln /> <ci> k </ci> </apply> <ci> j </ci> </apply> <apply> <power /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </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>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ZetaClassical", "[", RowBox[List["s_", ",", "a_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", RowBox[List["s", "-", "1"]]], "-", RowBox[List["PolyGamma", "[", "a", "]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["StieltjesGamma", "[", "1", "]"]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List["(", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["Log", "[", RowBox[List["k", "+", "a", "-", "1"]], "]"]], "j"], RowBox[List["k", "+", "a", "-", "1"]]], "-", FractionBox[SuperscriptBox[RowBox[List["Log", "[", "k", "]"]], "j"], "k"]]], ")"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["s", "-", "1"]], ")"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["StieltjesGamma", "[", "2", "]"]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List["(", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["Log", "[", RowBox[List["k", "+", "a", "-", "1"]], "]"]], "j"], RowBox[List["k", "+", "a", "-", "1"]]], "-", FractionBox[SuperscriptBox[RowBox[List["Log", "[", "k", "]"]], "j"], "k"]]], ")"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["s", "-", "1"]], ")"]], "2"]]], "+", SuperscriptBox[RowBox[List["O", "[", RowBox[List["s", "-", "1"]], "]"]], "3"]]]]]]]










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





2007-05-02





© 1998-2014 Wolfram Research, Inc.