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 Zeta

 http://functions.wolfram.com/10.02.06.0014.01

 Input Form

 Derivative[1, 0][Zeta][-n, a] \[Proportional] (BernoulliB[n + 1, a] Log[a])/(n + 1) - ((1/2) a^n (n + 1) + BernoulliB[n + 1, a])/(n + 1)^2 + (-1)^n n! Sum[(a^(-1 - k) BernoulliB[2 + k + n])/Pochhammer[1 + k, 2 + n], {k, 0, Infinity}] - (1/(n + 1)) Sum[BernoulliB[k] Sum[((-1)^j a^(1 - k + n) Binomial[n + 1, k])/(k - j), {j, 0, k - 1}], {k, 2, n + 1}] /; Element[n, Integers] && n > 0 && Inequality[-(Pi/2), Less, Arg[a], LessEqual, Pi/2] && (Re[a] -> Infinity)

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["Zeta", TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List[RowBox[List["-", "n"]], ",", "a"]], "]"]], "\[Proportional]", RowBox[List[FractionBox[RowBox[List[RowBox[List["BernoulliB", "[", RowBox[List[RowBox[List["n", "+", "1"]], ",", "a"]], "]"]], " ", RowBox[List["Log", "[", "a", "]"]]]], RowBox[List["n", "+", "1"]]], "-", FractionBox[RowBox[List[RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["a", "n"], " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]]]], "+", RowBox[List["BernoulliB", "[", RowBox[List[RowBox[List["n", "+", "1"]], ",", "a"]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], "2"]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["n", "!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], InterpretationBox["\[Infinity]", DirectedInfinity[1]]], FractionBox[RowBox[List[SuperscriptBox["a", RowBox[List[RowBox[List["-", "1"]], "-", "k"]]], " ", RowBox[List["BernoulliB", "[", RowBox[List["2", "+", "k", "+", "n"]], "]"]]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "k"]], ",", RowBox[List["2", "+", "n"]]]], "]"]]]]]]], "-", RowBox[List[FractionBox["1", RowBox[List["n", "+", "1"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "2"]], RowBox[List["n", "+", "1"]]], RowBox[List[RowBox[List["BernoulliB", "[", "k", "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["k", "-", "1"]]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", SuperscriptBox["a", RowBox[List["1", "-", "k", "+", "n"]]], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "+", "1"]], ",", "k"]], "]"]]]], RowBox[List["k", "-", "j"]]]]]]]]]]]]]]], "/;", "\[IndentingNewLine]", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]], "\[And]", RowBox[List[RowBox[List["-", FractionBox["\[Pi]", "2"]]], "<", RowBox[List["Arg", "[", "a", "]"]], "\[LessEqual]", FractionBox["\[Pi]", "2"]]], "\[And]", RowBox[List["(", RowBox[List[RowBox[List["Re", "[", "a", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]

 MathML Form

 ζ ( 1 , 0 ) TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative] ( - n , a ) B TagBox["B", BernoulliB] n + 1 ( a ) log ( a ) n + 1 + ( - 1 ) n n ! k = 0 a - k - 1 B TagBox["B", BernoulliB] k + n + 2 ( k + 1 ) n + 2 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], RowBox[List["n", "+", "2"]]], Pochhammer] - 1 n + 1 k = 2 n + 1 B TagBox["B", BernoulliB] k j = 0 k - 1 ( n + 1 k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["n", "+", "1"]], Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( - 1 ) j a - k + n + 1 k - j - 1 2 ( n + 1 ) a n + B TagBox["B", BernoulliB] n + 1 ( a ) ( n + 1 ) 2 /; n + - π 2 < arg ( a ) π 2 ( Re ( a ) "\[Rule]" ) Condition Proportional 1 0 Zeta -1 n a BernoulliB n 1 a a n 1 -1 -1 n n k 0 a -1 k -1 BernoulliB k n 2 Pochhammer k 1 n 2 -1 -1 1 n 1 -1 k 2 n 1 BernoulliB k j 0 k -1 Binomial n 1 k -1 j a -1 k n 1 k -1 j -1 -1 1 2 n 1 a n BernoulliB n 1 a n 1 2 -1 n SuperPlus Inequality -1 2 -1 a 2 -1 Rule a [/itex]

 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["-", "n_"]], ",", "a_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["BernoulliB", "[", RowBox[List[RowBox[List["n", "+", "1"]], ",", "a"]], "]"]], " ", RowBox[List["Log", "[", "a", "]"]]]], RowBox[List["n", "+", "1"]]], "-", FractionBox[RowBox[List[RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["a", "n"], " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]]]], "+", RowBox[List["BernoulliB", "[", RowBox[List[RowBox[List["n", "+", "1"]], ",", "a"]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], "2"]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["n", "!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox["a", RowBox[List[RowBox[List["-", "1"]], "-", "k"]]], " ", RowBox[List["BernoulliB", "[", RowBox[List["2", "+", "k", "+", "n"]], "]"]]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "k"]], ",", RowBox[List["2", "+", "n"]]]], "]"]]]]]]], "-", FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "2"]], RowBox[List["n", "+", "1"]]], RowBox[List[RowBox[List["BernoulliB", "[", "k", "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["k", "-", "1"]]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", SuperscriptBox["a", RowBox[List["1", "-", "k", "+", "n"]]], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "+", "1"]], ",", "k"]], "]"]]]], RowBox[List["k", "-", "j"]]]]]]]]], RowBox[List["n", "+", "1"]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]], "&&", RowBox[List[RowBox[List["-", FractionBox["\[Pi]", "2"]]], "<", RowBox[List["Arg", "[", "a", "]"]], "\[LessEqual]", FractionBox["\[Pi]", "2"]]], "&&", RowBox[List["(", RowBox[List[RowBox[List["Re", "[", "a", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]]]

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

 2002-12-18