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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 on branch cuts > For zeta(s,a)





http://functions.wolfram.com/10.02.06.0046.01









  


  










Input Form





Zeta[s, a] \[Proportional] (1 + O[a - Subscript[a, 0]])/ (I^s E^(I Pi s Floor[(-(1/Pi)) (Arg[(a + n)/(Subscript[a, 0] + n)] + Arg[(-I) (Subscript[a, 0] + n)])]) ((-I) (Subscript[a, 0] + n))^ s) + (Sum[1/(-Subscript[a, 0] - k)^s, {k, 0, n - 1}] + Zeta[s, 1 + Subscript[a, 0] + n]) (1 + O[a - Subscript[a, 0]]) /; Re[Subscript[a, 0]] == -n && Element[n, Integers] && n >= 0










Standard Form





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MathML Form







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</ci> </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[SuperscriptBox["\[ImaginaryI]", RowBox[List["-", "s"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[Pi]", " ", "s", " ", RowBox[List["Floor", "[", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["Arg", "[", FractionBox[RowBox[List["a", "+", "n"]], RowBox[List[SubscriptBox["aa", "0"], "+", "n"]]], "]"]], "+", RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List[SubscriptBox["aa", "0"], "+", "n"]], ")"]]]], "]"]]]], "\[Pi]"]]], "]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List[SubscriptBox["aa", "0"], "+", "n"]], ")"]]]], ")"]], RowBox[List["-", "s"]]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", RowBox[List["a", "-", SubscriptBox["aa", "0"]]], "]"]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", SubscriptBox["aa", "0"]]], "-", "k"]], ")"]], "s"]]]], "+", RowBox[List["Zeta", "[", RowBox[List["s", ",", RowBox[List["1", "+", SubscriptBox["aa", "0"], "+", "n"]]]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", RowBox[List["a", "-", SubscriptBox["aa", "0"]]], "]"]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", SubscriptBox["aa", "0"], "]"]], "\[Equal]", RowBox[List["-", "n"]]]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]










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





2007-05-02





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