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SphericalBesselJ






Mathematica Notation

Traditional Notation









Bessel-Type Functions > SphericalBesselJ[nu,z] > Series representations > Asymptotic series expansions > Expansions for any z in trigonometric form > Using trigonometric functions with branch cut-containing arguments





http://functions.wolfram.com/03.21.06.0064.01









  


  










Input Form





SphericalBesselJ[\[Nu], z] \[Proportional] z^\[Nu] (z^2)^(-(1/2) - \[Nu]/2) (Sin[Sqrt[z^2] - (Pi \[Nu])/2] (1 - ((-1 + \[Nu]) \[Nu] (1 + \[Nu]) (2 + \[Nu]))/(8 z^2) + ((-3 + \[Nu]) (-2 + \[Nu]) (-1 + \[Nu]) \[Nu] (1 + \[Nu]) (2 + \[Nu]) (3 + \[Nu]) (4 + \[Nu]))/(384 z^4) + \[Ellipsis]) + ((\[Nu] (1 + \[Nu]))/(2 Sqrt[z^2])) Cos[Sqrt[z^2] - (Pi \[Nu])/2] (1 - ((-2 + \[Nu]) (-1 + \[Nu]) (2 + \[Nu]) (3 + \[Nu]))/(24 z^2) + ((-4 + \[Nu]) (-3 + \[Nu]) (-2 + \[Nu]) (-1 + \[Nu]) (2 + \[Nu]) (3 + \[Nu]) (4 + \[Nu]) (5 + \[Nu]))/(1920 z^4) + \[Ellipsis])) /; (Abs[z] -> Infinity)










Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["z", "\[Nu]"], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "2"], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", FractionBox["\[Nu]", "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Sin", "[", RowBox[List[SqrtBox[SuperscriptBox["z", "2"]], "-", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ")"]], " ", "\[Nu]", " ", RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "\[Nu]"]], ")"]]]], RowBox[List["8", " ", SuperscriptBox["z", "2"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ")"]], " ", "\[Nu]", " ", RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["4", "+", "\[Nu]"]], ")"]]]], RowBox[List["384", " ", SuperscriptBox["z", "4"]]]], "+", "\[Ellipsis]"]], ")"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", " ", RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]]]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List[SqrtBox[SuperscriptBox["z", "2"]], "-", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", "\[Nu]"]], ")"]]]], RowBox[List["24", " ", SuperscriptBox["z", "2"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "4"]], "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["4", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["5", "+", "\[Nu]"]], ")"]]]], RowBox[List["1920", " ", SuperscriptBox["z", "4"]]]], "+", "\[Ellipsis]"]], ")"]]]], RowBox[List["2", " ", SqrtBox[SuperscriptBox["z", "2"]]]]]]], ")"]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]










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





2007-05-02