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variants of this functions
Zeta






Mathematica Notation

Traditional Notation









Zeta Functions and Polylogarithms > Zeta[s,a] > Differentiation > Low-order differentiation > With respect to s > For zeta(s,a) > Derivatives at negarive integer points





http://functions.wolfram.com/10.02.20.0025.01









  


  










Input Form





Derivative[1, 0][Zeta][-2, a] == 2 PolyGamma[-3, -1 + a] + (-1 + a)^2 Log[-1 + a] + (1/(12 Pi^2)) (2 (-1 + a) Pi^2 (2 a^2 EulerGamma - a (9 + EulerGamma + Log[8] + 3 Log[Pi]) + 3 (3 - 4 Log[Glaisher] + Log[2 Pi])) - 3 Zeta[3]) + (-(3/2) + EulerGamma) Zeta[-2, a] + Sum[(1/2) (a + k)^2 (2 Log[a + k] - Log[(a + k)^2]), {k, 0, Floor[1 - Re[a]]}]










Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox["Zeta", TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List[RowBox[List["-", "2"]], ",", "a_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["2", " ", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "3"]], ",", RowBox[List[RowBox[List["-", "1"]], "+", "a"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "a"]], ")"]], "2"], " ", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "1"]], "+", "a"]], "]"]]]], "+", FractionBox[RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "a"]], ")"]], " ", SuperscriptBox["\[Pi]", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", "EulerGamma"]], "-", RowBox[List["a", " ", RowBox[List["(", RowBox[List["9", "+", "EulerGamma", "+", RowBox[List["Log", "[", "8", "]"]], "+", RowBox[List["3", " ", RowBox[List["Log", "[", "\[Pi]", "]"]]]]]], ")"]]]], "+", RowBox[List["3", " ", RowBox[List["(", RowBox[List["3", "-", RowBox[List["4", " ", RowBox[List["Log", "[", "Glaisher", "]"]]]], "+", RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]]]], ")"]]]]]], ")"]]]], "-", RowBox[List["3", " ", RowBox[List["Zeta", "[", "3", "]"]]]]]], RowBox[List["12", " ", SuperscriptBox["\[Pi]", "2"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["3", "2"]]], "+", "EulerGamma"]], ")"]], " ", RowBox[List["Zeta", "[", RowBox[List[RowBox[List["-", "2"]], ",", "a"]], "]"]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List["1", "-", RowBox[List["Re", "[", "a", "]"]]]], "]"]]], RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "k"]], ")"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["Log", "[", RowBox[List["a", "+", "k"]], "]"]]]], "-", RowBox[List["Log", "[", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "k"]], ")"]], "2"], "]"]]]], ")"]]]]]]]]]]]]










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





2007-05-02





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