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 SphericalBesselJ

 http://functions.wolfram.com/03.21.20.0004.01

 Input Form

 Derivative[1, 0][SphericalBesselJ][-n, z] - (-(((-1)^n 2^(1 - n))/(z^n (n - 1)!))) (2 z Sum[(-4)^k z^(2 k) Binomial[-1 + n, 1 + 2 k] (-3 - 2 k + 2 n)! (CosIntegral[2 z] Sin[z] + (PolyGamma[3/2 + k] - PolyGamma[3/2 + k - n]) Sin[z] - Cos[z] SinIntegral[2 z]), {k, 0, -1 + Floor[n/2]}] + Sum[(-4)^k z^(2 k) Binomial[-1 + n, 2 k] (-2 - 2 k + 2 n)! (Cos[z] CosIntegral[2 z] + Cos[z] (PolyGamma[1/2 + k] - PolyGamma[3/2 + k - n]) + Sin[z] SinIntegral[2 z]), {k, 0, Floor[(n - 1)/2]}]) /; Element[n, Integers] && n > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[RowBox[List["Derivative", "[", RowBox[List["1", ",", "0"]], "]"]], "[", "SphericalBesselJ", "]"]], "[", RowBox[List[RowBox[List["-", "n"]], ",", "z"]], "]"]], "-", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SuperscriptBox["2", RowBox[List["1", "-", "n"]]], " ", SuperscriptBox["z", RowBox[List["-", "n"]]], " "]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]]]]], RowBox[List["(", RowBox[List[RowBox[List["2", " ", "z", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "4"]], ")"]], "k"], " ", SuperscriptBox["z", RowBox[List["2", " ", "k"]]], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ",", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]]]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "-", RowBox[List["2", " ", "k"]], "+", RowBox[List["2", " ", "n"]]]], ")"]], "!"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["CosIntegral", "[", RowBox[List["2", " ", "z"]], "]"]], " ", RowBox[List["Sin", "[", "z", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["3", "2"], "+", "k"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["3", "2"], "+", "k", "-", "n"]], "]"]]]], ")"]], " ", RowBox[List["Sin", "[", "z", "]"]]]], "-", RowBox[List[RowBox[List["Cos", "[", "z", "]"]], " ", RowBox[List["SinIntegral", "[", RowBox[List["2", " ", "z"]], "]"]]]]]], ")"]]]]]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "1"]], "2"], "]"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "4"]], ")"]], "k"], " ", SuperscriptBox["z", RowBox[List["2", " ", "k"]]], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ",", RowBox[List["2", " ", "k"]]]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "-", RowBox[List["2", " ", "k"]], "+", RowBox[List["2", " ", "n"]]]], ")"]], "!"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Cos", "[", "z", "]"]], " ", RowBox[List["CosIntegral", "[", RowBox[List["2", " ", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["Cos", "[", "z", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "+", "k"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["3", "2"], "+", "k", "-", "n"]], "]"]]]], ")"]]]], "+", RowBox[List[RowBox[List["Sin", "[", "z", "]"]], " ", RowBox[List["SinIntegral", "[", RowBox[List["2", " ", "z"]], "]"]]]]]], ")"]]]]]]]], ")"]]]], ")"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]]]]]]]]

 MathML Form

 j TagBox["j", BesselJ] - n ( 1 , 0 ) TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative] ( z ) - ( - 1 ) n 2 1 - n z - n ( n - 1 ) ! ( k = 0 n - 1 2 ( - 4 ) k z 2 k ( n - 1 2 k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["n", "-", "1"]], 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]] ( 2 n - 2 k - 2 ) ! ( cos ( z ) Ci ( 2 z ) + cos ( z ) ( ψ TagBox["\[Psi]", PolyGamma] ( k + 1 2 ) - ψ TagBox["\[Psi]", PolyGamma] ( k - n + 3 2 ) ) + sin ( z ) Si ( 2 z ) ) + 2 z k = 0 n 2 - 1 ( - 4 ) k z 2 k ( n - 1 2 k + 1 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["n", "-", "1"]], 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]] ( 2 n - 2 k - 3 ) ! ( Ci ( 2 z ) sin ( z ) + ( ψ TagBox["\[Psi]", PolyGamma] ( k + 3 2 ) - ψ TagBox["\[Psi]", PolyGamma] ( k - n + 3 2 ) ) sin ( z ) - cos ( z ) Si ( 2 z ) ) ) /; n + TagBox[SuperscriptBox["\[DoubleStruckCapitalN]", "+"], Function[Integers]] Condition 1 0 Subscript BesselJ j -1 n z -1 -1 n 2 1 -1 n z -1 n n -1 -1 k 0 n -1 2 -1 -4 k z 2 k Binomial n -1 2 k 2 n -1 2 k -2 z CosIntegral 2 z z PolyGamma k 1 2 -1 PolyGamma k -1 n 3 2 z SinIntegral 2 z 2 z k 0 n 2 -1 -1 -4 k z 2 k Binomial n -1 2 k 1 2 n -1 2 k -3 CosIntegral 2 z z PolyGamma k 3 2 -1 PolyGamma k -1 n 3 2 z -1 z SinIntegral 2 z n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["SphericalBesselJ", TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List[RowBox[List["-", "n"]], ",", "z"]], "]"]], "-", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SuperscriptBox["2", RowBox[List["1", "-", "n"]]], " ", SuperscriptBox["z", RowBox[List["-", "n"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "z", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "4"]], ")"]], "k"], " ", SuperscriptBox["z", RowBox[List["2", " ", "k"]]], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ",", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]]]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "-", RowBox[List["2", " ", "k"]], "+", RowBox[List["2", " ", "n"]]]], ")"]], "!"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["CosIntegral", "[", RowBox[List["2", " ", "z"]], "]"]], " ", RowBox[List["Sin", "[", "z", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["3", "2"], "+", "k"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["3", "2"], "+", "k", "-", "n"]], "]"]]]], ")"]], " ", RowBox[List["Sin", "[", "z", "]"]]]], "-", RowBox[List[RowBox[List["Cos", "[", "z", "]"]], " ", RowBox[List["SinIntegral", "[", RowBox[List["2", " ", "z"]], "]"]]]]]], ")"]]]]]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "1"]], "2"], "]"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "4"]], ")"]], "k"], " ", SuperscriptBox["z", RowBox[List["2", " ", "k"]]], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ",", RowBox[List["2", " ", "k"]]]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "-", RowBox[List["2", " ", "k"]], "+", RowBox[List["2", " ", "n"]]]], ")"]], "!"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Cos", "[", "z", "]"]], " ", RowBox[List["CosIntegral", "[", RowBox[List["2", " ", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["Cos", "[", "z", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "+", "k"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["3", "2"], "+", "k", "-", "n"]], "]"]]]], ")"]]]], "+", RowBox[List[RowBox[List["Sin", "[", "z", "]"]], " ", RowBox[List["SinIntegral", "[", RowBox[List["2", " ", "z"]], "]"]]]]]], ")"]]]]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]

 Contributed by

 Brychkov Yu.A. (2005)

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

 2007-05-02