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 SphericalBesselJ

 http://functions.wolfram.com/03.21.06.0001.01

 Input Form

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

 Standard Form

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

 MathML Form

 j ν ( z ) j n ( z ) + 2 ( 2 z ) - n - 1 n ! ( 2 z k = 0 n - 1 2 ( - 1 ) k 2 2 k ( 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]] ( - 2 k + 2 n - 1 ) ! ( - cos ( z ) Ci ( 2 z ) + cos ( z ) ( ψ TagBox["\[Psi]", PolyGamma] ( k + 3 2 ) - ψ TagBox["\[Psi]", PolyGamma] ( k - n + 1 2 ) ) - sin ( z ) Si ( 2 z ) ) z 2 k - k = 0 n 2 ( - 1 ) k 2 2 k ( 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]] ( 2 n - 2 k ) ! ( - Ci ( 2 z ) sin ( z ) + ( ψ TagBox["\[Psi]", PolyGamma] ( k + 1 2 ) - ψ TagBox["\[Psi]", PolyGamma] ( k - n + 1 2 ) ) sin ( z ) + cos ( z ) Si ( 2 z ) ) z 2 k ) ( ν - n ) + /; ( ν "\[Rule]" n ) n TagBox["\[DoubleStruckCapitalN]", Function[Integers]] Condition Proportional SphericalBesselJ ν z SphericalBesselJ n z 2 2 z -1 n -1 n -1 2 z k 0 n -1 2 -1 -1 k 2 2 k Binomial n 2 k 1 -2 k 2 n -1 -1 z CosIntegral 2 z z PolyGamma k 3 2 -1 PolyGamma k -1 n 1 2 -1 z SinIntegral 2 z z 2 k -1 k 0 n 2 -1 -1 k 2 2 k Binomial n 2 k 2 n -1 2 k -1 CosIntegral 2 z z PolyGamma k 1 2 -1 PolyGamma k -1 n 1 2 z z SinIntegral 2 z z 2 k ν -1 n Rule ν n n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["SphericalBesselJ", "[", RowBox[List["n", ",", "z"]], "]"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "z"]], ")"]], RowBox[List[RowBox[List["-", "n"]], "-", "1"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["2", RowBox[List["2", " ", "k"]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", RowBox[List["2", " ", "k"]]]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "-", RowBox[List["2", " ", "k"]]]], ")"]], "!"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "+", FractionBox["1", "2"]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["k", "-", "n", "+", FractionBox["1", "2"]]], "]"]]]], ")"]], " ", RowBox[List["Sin", "[", "z", "]"]]]], "-", RowBox[List[RowBox[List["Sin", "[", "z", "]"]], " ", RowBox[List["CosIntegral", "[", RowBox[List["2", " ", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["Cos", "[", "z", "]"]], " ", RowBox[List["SinIntegral", "[", RowBox[List["2", " ", "z"]], "]"]]]]]], ")"]], " ", SuperscriptBox["z", RowBox[List["2", " ", "k"]]]]]]]]], "+", 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["-", "1"]], ")"]], "k"], " ", SuperscriptBox["2", RowBox[List["2", " ", "k"]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "-", RowBox[List["2", " ", "k"]], "-", "1"]], ")"]], "!"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "+", FractionBox["3", "2"]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["k", "-", "n", "+", FractionBox["1", "2"]]], "]"]]]], ")"]], " ", RowBox[List["Cos", "[", "z", "]"]]]], "-", RowBox[List[RowBox[List["Sin", "[", "z", "]"]], " ", RowBox[List["SinIntegral", "[", RowBox[List["2", " ", "z"]], "]"]]]], "-", RowBox[List[RowBox[List["Cos", "[", "z", "]"]], " ", RowBox[List["CosIntegral", "[", RowBox[List["2", " ", "z"]], "]"]]]]]], ")"]], " ", SuperscriptBox["z", RowBox[List["2", " ", "k"]]]]]]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["\[Nu]", "-", "n"]], ")"]]]], RowBox[List["n", "!"]]], "+", "\[Ellipsis]"]], "/;", RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", "\[Rule]", "n"]], ")"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2007-05-02