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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==a0/;a0!=-n > For zeta(s,a)





http://functions.wolfram.com/10.02.06.0031.01









  


  








Input Form





Zeta[s, a] \[Proportional] Zeta[s, Subscript[a, 0]] - Zeta[s, 1 + Floor[-Re[Subscript[a, 0]]] + Subscript[a, 0]] + Zeta[s, Max[0, 1 + Floor[-Re[Subscript[a, 0]]]] + Subscript[a, 0]] - s (Zeta[1 + s, Subscript[a, 0]] - Zeta[1 + s, 1 + Floor[-Re[Subscript[a, 0]]] + Subscript[a, 0]] + Zeta[1 + s, Max[0, 1 + Floor[-Re[Subscript[a, 0]]]] + Subscript[a, 0]]) (a - Subscript[a, 0]) + (1/2) (s + 1) s (Zeta[-2 (-1 - s/2), Subscript[a, 0]] - Zeta[-2 (-1 - s/2), 1 + Floor[-Re[Subscript[a, 0]]] + Subscript[a, 0]] + Zeta[2 + s, Max[0, 1 + Floor[-Re[Subscript[a, 0]]]] + Subscript[a, 0]]) (a - Subscript[a, 0])^2 + \[Ellipsis] /; (a -> Subscript[a, 0])








Standard Form





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MathML Form







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&quot;)&quot;]]]], &quot;\[RightFloor]&quot;]], &quot;+&quot;, &quot;1&quot;]]]], &quot;)&quot;]], &quot;+&quot;, SubscriptBox[&quot;a&quot;, &quot;0&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> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mo> &#8230; </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <msub> <mi> a </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </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> <ci> Zeta </ci> <ci> s </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Zeta </ci> <ci> s </ci> <apply> <plus /> <apply> <floor /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <real /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <ci> Zeta </ci> <ci> s </ci> <apply> <plus /> <apply> <max /> <cn type='integer'> 0 </cn> <apply> <plus /> <apply> <floor /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <real /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> s </ci> <apply> <plus /> <apply> <ci> Zeta </ci> <apply> <plus /> <ci> s </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Zeta </ci> <apply> <plus /> <ci> s </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <floor /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <real /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <ci> Zeta </ci> <apply> <plus /> <ci> s </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <max /> <cn type='integer'> 0 </cn> <apply> <plus /> <apply> <floor /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <real /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </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> <plus /> <apply> <ci> Zeta </ci> <apply> <times /> <cn type='integer'> -2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <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> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Zeta </ci> <apply> <times /> <cn type='integer'> -2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <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> <plus /> <apply> <floor /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <real /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <ci> Zeta </ci> <apply> <plus /> <ci> s </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <max /> <cn type='integer'> 0 </cn> <apply> <plus /> <apply> <floor /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <real /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> <apply> <ci> Rule </ci> <ci> a </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </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["s", ",", SubscriptBox["aa", "0"]]], "]"]], "-", RowBox[List["Zeta", "[", RowBox[List["s", ",", RowBox[List["1", "+", RowBox[List["Floor", "[", RowBox[List["-", RowBox[List["Re", "[", SubscriptBox["aa", "0"], "]"]]]], "]"]], "+", SubscriptBox["aa", "0"]]]]], "]"]], "+", RowBox[List["Zeta", "[", RowBox[List["s", ",", RowBox[List[RowBox[List["Max", "[", RowBox[List["0", ",", RowBox[List["1", "+", RowBox[List["Floor", "[", RowBox[List["-", RowBox[List["Re", "[", SubscriptBox["aa", "0"], "]"]]]], "]"]]]]]], "]"]], "+", SubscriptBox["aa", "0"]]]]], "]"]], "-", RowBox[List["s", " ", RowBox[List["(", RowBox[List[RowBox[List["Zeta", "[", RowBox[List[RowBox[List["1", "+", "s"]], ",", SubscriptBox["aa", "0"]]], "]"]], "-", RowBox[List["Zeta", "[", RowBox[List[RowBox[List["1", "+", "s"]], ",", RowBox[List["1", "+", RowBox[List["Floor", "[", RowBox[List["-", RowBox[List["Re", "[", SubscriptBox["aa", "0"], "]"]]]], "]"]], "+", SubscriptBox["aa", "0"]]]]], "]"]], "+", RowBox[List["Zeta", "[", RowBox[List[RowBox[List["1", "+", "s"]], ",", RowBox[List[RowBox[List["Max", "[", RowBox[List["0", ",", RowBox[List["1", "+", RowBox[List["Floor", "[", RowBox[List["-", RowBox[List["Re", "[", SubscriptBox["aa", "0"], "]"]]]], "]"]]]]]], "]"]], "+", SubscriptBox["aa", "0"]]]]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["a", "-", SubscriptBox["aa", "0"]]], ")"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["s", "+", "1"]], ")"]], " ", "s", " ", RowBox[List["(", RowBox[List[RowBox[List["Zeta", "[", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", FractionBox["s", "2"]]], ")"]]]], ",", SubscriptBox["aa", "0"]]], "]"]], "-", RowBox[List["Zeta", "[", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", FractionBox["s", "2"]]], ")"]]]], ",", RowBox[List["1", "+", RowBox[List["Floor", "[", RowBox[List["-", RowBox[List["Re", "[", SubscriptBox["aa", "0"], "]"]]]], "]"]], "+", SubscriptBox["aa", "0"]]]]], "]"]], "+", RowBox[List["Zeta", "[", RowBox[List[RowBox[List["2", "+", "s"]], ",", RowBox[List[RowBox[List["Max", "[", RowBox[List["0", ",", RowBox[List["1", "+", RowBox[List["Floor", "[", RowBox[List["-", RowBox[List["Re", "[", SubscriptBox["aa", "0"], "]"]]]], "]"]]]]]], "]"]], "+", SubscriptBox["aa", "0"]]]]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", SubscriptBox["aa", "0"]]], ")"]], "2"]]], "+", "\[Ellipsis]"]], "/;", RowBox[List["(", RowBox[List["a", "\[Rule]", SubscriptBox["aa", "0"]]], ")"]]]]]]]]


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





2007-05-02