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 KelvinBei

 http://functions.wolfram.com/03.17.20.0017.01

 Input Form

 D[KelvinBei[\[Nu], z], {z, n}] == 2^(-1 - 3 (n/2)) (I - 1)^n Sum[(((1 + n) Binomial[n, 2 k])/(1 + 2 k)) ((1 + I^n) KelvinBei[4 k - n + \[Nu], z] + (-I + I^(n + 1)) KelvinBer[4 k - n + \[Nu], z]) + (1/z) (1 + I) Sqrt[2] (1 + 4 k - n + \[Nu]) Binomial[n, 1 + 2 k] ((1 - I^(n + 1)) KelvinBei[1 + 4 k - n + \[Nu], z] + (-I + I^n) KelvinBer[1 + 4 k - n + \[Nu], z]), {k, 0, Floor[n/2]}] /; Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "n"]], "}"]]], RowBox[List["KelvinBei", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "\[Equal]", "\n", "\t", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["3", RowBox[List["n", "/", "2"]]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", "-", "1"]], ")"]], "n"], RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List["n", "/", "2"]], "]"]]], RowBox[List["(", " ", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", RowBox[List["2", " ", "k"]]]], "]"]], " "]], RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ImaginaryI]", "n"]]], ")"]], " ", RowBox[List["KelvinBei", "[", RowBox[List[RowBox[List[RowBox[List["4", " ", "k"]], "-", "n", "+", "\[Nu]"]], ",", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], "+", SuperscriptBox["\[ImaginaryI]", RowBox[List["n", "+", "1"]]]]], ")"]], " ", RowBox[List["KelvinBer", "[", RowBox[List[RowBox[List[RowBox[List["4", " ", "k"]], "-", "n", "+", "\[Nu]"]], ",", "z"]], "]"]]]]]], ")"]]]], "+", RowBox[List[FractionBox["1", "z"], RowBox[List["(", RowBox[List["1", "+", "\[ImaginaryI]"]], ")"]], " ", SqrtBox["2"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["4", " ", "k"]], "-", "n", "+", "\[Nu]"]], ")"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["\[ImaginaryI]", RowBox[List["n", "+", "1"]]]]], ")"]], " ", RowBox[List["KelvinBei", "[", RowBox[List[RowBox[List["1", "+", RowBox[List["4", " ", "k"]], "-", "n", "+", "\[Nu]"]], ",", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], "+", SuperscriptBox["\[ImaginaryI]", "n"]]], ")"]], " ", RowBox[List["KelvinBer", "[", RowBox[List[RowBox[List["1", "+", RowBox[List["4", " ", "k"]], "-", "n", "+", "\[Nu]"]], ",", "z"]], "]"]]]]]], ")"]]]]]], ")"]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 n bei ν ( z ) z n 2 - 3 n 2 - 1 ( - 1 ) n k = 0 n 2 ( n + 1 2 k + 1 ( n 2 k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox[RowBox[List["2", " ", "k"]], Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] ( ( 1 + n ) bei 4 k - n + ν ( z ) + ( - + n + 1 ) ber 4 k - n + ν ( z ) ) + 2 ( 1 + ) ( 4 k - n + ν + 1 ) z ( n 2 k + 1 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox[RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] ( ( 1 - n + 1 ) bei 4 k - n + ν + 1 ( z ) + ( - + n ) ber 4 k - n + ν + 1 ( z ) ) ) /; n Condition z n KelvinBei ν z 2 -1 3 n 2 -1 -1 -1 n k 0 n 2 -1 n 1 2 k 1 -1 Binomial n 2 k 1 n KelvinBei 4 k -1 n ν z -1 n 1 KelvinBer 4 k -1 n ν z 2 1 2 1 4 k -1 n ν 1 z -1 Binomial n 2 k 1 1 -1 n 1 KelvinBei 4 k -1 n ν 1 z -1 n KelvinBer 4 k -1 n ν 1 z n [/itex]

 Rule Form

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

 2007-05-02