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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 s==s0/;s0!=1 > For zeta(s,a)





http://functions.wolfram.com/10.02.06.0021.01









  


  








Input Form





Zeta[s, a] \[Proportional] Zeta[Subscript[s, 0], a] + Derivative[1, 0][Zeta][Subscript[s, 0], a] (s - Subscript[s, 0]) + (1/2) Derivative[2, 0][Zeta][Subscript[s, 0], a] (s - Subscript[s, 0])^2 + \[Ellipsis] /; (s -> Subscript[s, 0]) && Subscript[s, 0] != 1








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> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <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> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <semantics> <msub> <mi> s </mi> <mn> 0 </mn> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[&quot;s&quot;, &quot;0&quot;], Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mrow> <msup> <mi> &#950; </mi> <semantics> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;1&quot;, &quot;,&quot;, &quot;0&quot;]], &quot;)&quot;]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mrow> <msub> <mi> s </mi> <mn> 0 </mn> </msub> <mo> , </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> s </mi> <mo> - </mo> <msub> <mi> s </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> &#950; </mi> <semantics> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;2&quot;, &quot;,&quot;, &quot;0&quot;]], &quot;)&quot;]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mrow> <msub> <mi> s </mi> <mn> 0 </mn> </msub> <mo> , </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> s </mi> <mo> - </mo> <msub> <mi> s </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mo> &#8230; </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> s </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <msub> <mi> s </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> s </mi> <mn> 0 </mn> </msub> <mo> &#8800; </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> Zeta </ci> <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> <ci> Zeta </ci> <apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> <apply> <ci> Subscript </ci> <ci> s </ci> <cn type='integer'> 0 </cn> </apply> </apply> <ci> a </ci> </apply> <apply> <times /> <apply> <apply> <partialdiff /> <list> <cn type='integer'> 1 </cn> <cn type='integer'> 0 </cn> </list> <ci> Zeta </ci> </apply> <apply> <ci> Subscript </ci> <ci> s </ci> <cn type='integer'> 0 </cn> </apply> <ci> a </ci> </apply> <apply> <plus /> <ci> s </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> s </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <apply> <partialdiff /> <list> <cn type='integer'> 2 </cn> <cn type='integer'> 0 </cn> </list> <ci> Zeta </ci> </apply> <apply> <ci> Subscript </ci> <ci> s </ci> <cn type='integer'> 0 </cn> </apply> <ci> a </ci> </apply> <apply> <power /> <apply> <plus /> <ci> s </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> s </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> <apply> <and /> <apply> <ci> Rule </ci> <ci> s </ci> <apply> <ci> Subscript </ci> <ci> s </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <neq /> <apply> <ci> Subscript </ci> <ci> s </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Zeta", "[", RowBox[List["s_", ",", "a_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["Zeta", "[", RowBox[List[SubscriptBox["ss", "0"], ",", "a"]], "]"]], "+", RowBox[List[RowBox[List[SuperscriptBox["Zeta", TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List[SubscriptBox["ss", "0"], ",", "a"]], "]"]], " ", RowBox[List["(", RowBox[List["s", "-", SubscriptBox["ss", "0"]]], ")"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List[SuperscriptBox["Zeta", TagBox[RowBox[List["(", RowBox[List["2", ",", "0"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List[SubscriptBox["ss", "0"], ",", "a"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["s", "-", SubscriptBox["ss", "0"]]], ")"]], "2"]]], "+", "\[Ellipsis]"]], "/;", RowBox[List[RowBox[List["(", RowBox[List["s", "\[Rule]", SubscriptBox["ss", "0"]]], ")"]], "&&", RowBox[List[SubscriptBox["ss", "0"], "\[NotEqual]", "1"]]]]]]]]]]


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





2007-05-02