html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 Zeta

 http://functions.wolfram.com/10.02.20.0024.01

 Input Form

 Derivative[1, 0][Zeta][-2, a] == (1/(12 Pi^2)) (2 (-9 + a (18 + 3 a (-6 + EulerGamma) + EulerGamma + 2 a^2 EulerGamma)) Pi^2 - 3 Zeta[3] + Pi^2 ((1 + 6 a (1 + a)) (-3 + 2 EulerGamma) Floor[-Re[a]] + 3 (1 + 2 a) (-3 + 2 EulerGamma) Floor[-Re[a]]^2 + (-6 + 4 EulerGamma) Floor[-Re[a]]^3 + 6 ((-1 + a) (Log[2] + 2 (-1 + a) Log[-1 + a] - 4 Log[Glaisher] + Log[Pi] - a Log[2 Pi]) + 4 PolyGamma[-3, -1 + a] + 2 Sum[(1/2) (a + k)^2 (2 Log[a + k] - Log[(a + k)^2]), {k, 0, Floor[-Re[a]]}] + (-3 + 2 EulerGamma) Zeta[-2, a + Max[0, Floor[1 - Re[a]]]])))

 Standard Form

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 MathML Form

 ζ ( 1 , 0 ) TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative] ( - 2 , a ) 1 12 π 2 ( 2 π 2 ( a ( 2 TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] a 2 + 3 ( - 6 + TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ) a + TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] + 18 ) - 9 ) - 3 ζ ( 3 ) TagBox[RowBox[List["\[Zeta]", "(", TagBox["3", Zeta, Rule[Editable, True]], ")"]], InterpretTemplate[Function[BoxForm`e\$, Zeta[BoxForm`e\$]]]] + π 2 ( ( - 6 + 4 TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ) - Re ( a ) 3 + 3 ( 2 a + 1 ) ( - 3 + 2 TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ) - Re ( a ) 2 + ( 6 a ( a + 1 ) + 1 ) ( - 3 + 2 TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ) - Re ( a ) + 6 ( ( a - 1 ) ( log ( 2 ) + 2 ( a - 1 ) log ( a - 1 ) - 4 log ( A TagBox["A", Function[List[], Glaisher]] ) + log ( π ) - a log ( 2 π ) ) + 4 ψ TagBox["\[Psi]", PolyGamma] ( - 3 ) ( a - 1 ) + 2 k = 0 - Re ( a ) 1 2 ( a + k ) 2 ( 2 log ( a + k ) - log ( ( a + k ) 2 ) ) + ( - 3 + 2 TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ) ζ ( - 2 , a + max ( 0 , 1 - Re ( a ) ) ) TagBox[RowBox[List["\[Zeta]", "(", RowBox[List[TagBox[RowBox[List["-", "2"]], Zeta, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "+", RowBox[List["max", "(", RowBox[List["0", ",", RowBox[List["\[LeftFloor]", RowBox[List["1", "-", RowBox[List["Re", "(", "a", ")"]]]], "\[RightFloor]"]]]], ")"]]]], Zeta, Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[List[ZetaDump`e1\$, ZetaDump`e2\$], Zeta[ZetaDump`e1\$, ZetaDump`e2\$]]]] ) ) ) 1 0 Zeta -2 a 1 12 2 -1 2 2 a 2 a 2 3 -6 a 18 -9 -1 3 Zeta 3 2 -6 4 -1 a 3 3 2 a 1 -3 2 -1 a 2 6 a a 1 1 -3 2 -1 a 6 a -1 2 2 a -1 a -1 -1 4 Glaisher -1 a 2 4 PolyGamma -3 a -1 2 k 0 -1 a 1 2 a k 2 2 a k -1 a k 2 -3 2 Zeta -2 a 0 1 -1 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["-", "2"]], ",", "a_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "9"]], "+", RowBox[List["a", " ", RowBox[List["(", RowBox[List["18", "+", RowBox[List["3", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "6"]], "+", "EulerGamma"]], ")"]]]], "+", "EulerGamma", "+", RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", "EulerGamma"]]]], ")"]]]]]], ")"]], " ", SuperscriptBox["\[Pi]", "2"]]], "-", RowBox[List["3", " ", RowBox[List["Zeta", "[", "3", "]"]]]], "+", RowBox[List[SuperscriptBox["\[Pi]", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["6", " ", "a", " ", RowBox[List["(", RowBox[List["1", "+", "a"]], ")"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", RowBox[List["2", " ", "EulerGamma"]]]], ")"]], " ", RowBox[List["Floor", "[", RowBox[List["-", RowBox[List["Re", "[", "a", "]"]]]], "]"]]]], "+", RowBox[List["3", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "a"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", RowBox[List["2", " ", "EulerGamma"]]]], ")"]], " ", SuperscriptBox[RowBox[List["Floor", "[", RowBox[List["-", RowBox[List["Re", "[", "a", "]"]]]], "]"]], "2"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "6"]], "+", RowBox[List["4", " ", "EulerGamma"]]]], ")"]], " ", SuperscriptBox[RowBox[List["Floor", "[", RowBox[List["-", RowBox[List["Re", "[", "a", "]"]]]], "]"]], "3"]]], "+", RowBox[List["6", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "a"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "2", "]"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "a"]], ")"]], " ", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "1"]], "+", "a"]], "]"]]]], "-", RowBox[List["4", " ", RowBox[List["Log", "[", "Glaisher", "]"]]]], "+", RowBox[List["Log", "[", "\[Pi]", "]"]], "-", RowBox[List["a", " ", RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]]]]]], ")"]]]], "+", RowBox[List["4", " ", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "3"]], ",", RowBox[List[RowBox[List["-", "1"]], "+", "a"]]]], "]"]]]], "+", RowBox[List["2", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List["-", 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"], "]"]]]], ")"]]]]]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", RowBox[List["2", " ", "EulerGamma"]]]], ")"]], " ", RowBox[List["Zeta", "[", RowBox[List[RowBox[List["-", "2"]], ",", RowBox[List["a", "+", RowBox[List["Max", "[", RowBox[List["0", ",", RowBox[List["Floor", "[", RowBox[List["1", "-", RowBox[List["Re", "[", "a", "]"]]]], "]"]]]], "]"]]]]]], "]"]]]]]], ")"]]]]]], ")"]]]]]], RowBox[List["12", " ", SuperscriptBox["\[Pi]", "2"]]]]]]]]

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

 2007-05-02