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 SphericalBesselJ

 http://functions.wolfram.com/03.21.06.0054.01

 Input Form

 SphericalBesselJ[\[Nu], z] \[Proportional] ((I (-1)^\[Nu] Sqrt[-z])/z^(3/2)) (((\[Nu] (1 + \[Nu]))/(2 z)) Cos[z + (Pi \[Nu])/2] HypergeometricPFQ[{1/2 - \[Nu]/2, 1 - \[Nu]/2, 1 + \[Nu]/2, 3/2 + \[Nu]/2}, {3/2}, -(1/z^2)] + Sin[z + (Pi \[Nu])/2] HypergeometricPFQ[{1/2 - \[Nu]/2, 1/2 + \[Nu]/2, 1 + \[Nu]/2, -(\[Nu]/2)}, {1/2}, -(1/z^2)]) /; Inequality[0, Less, Arg[z], LessEqual, Pi] && (Abs[z] -> Infinity)

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "\[Nu]"], " ", SqrtBox[RowBox[List["-", "z"]]]]], SuperscriptBox["z", RowBox[List["3", "/", "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox[RowBox[List["\[Nu]", " ", RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]]]], RowBox[List["2", "z"]]], " ", RowBox[List["Cos", "[", RowBox[List["z", "+", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"]]], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", FractionBox["\[Nu]", "2"]]], ",", RowBox[List["1", "-", FractionBox["\[Nu]", "2"]]], ",", RowBox[List["1", "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["3", "2"], "+", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", FractionBox["3", "2"], "}"]], ",", RowBox[List["-", FractionBox["1", SuperscriptBox["z", "2"]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["Sin", "[", RowBox[List["z", "+", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"]]], "]"]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["1", "2"], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List["1", "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List["-", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", FractionBox["1", "2"], "}"]], ",", RowBox[List["-", FractionBox["1", SuperscriptBox["z", "2"]]]]]], "]"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["0", "<", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", "\[Pi]"]], "\[And]", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]

 MathML Form

 j ν ( z ) ( - 1 ) ν - z z 3 / 2 ( ν ( ν + 1 ) 2 z cos ( z + π ν 2 ) 4 F 1 ( 1 2 - ν 2 , 1 - ν 2 , ν 2 + 1 , ν 2 + 3 2 ; 3 2 ; - 1 z 2 ) TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "4"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox["1", "2"], "-", FractionBox["\[Nu]", "2"]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["1", "-", FractionBox["\[Nu]", "2"]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", FractionBox["3", "2"]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[TagBox[FractionBox["3", "2"], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[RowBox[List["-", FractionBox["1", SuperscriptBox["z", "2"]]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]] + sin ( z + π ν 2 ) 4 F 1 ( 1 2 - ν 2 , ν 2 + 1 2 , ν 2 + 1 , - ν 2 ; 1 2 ; - 1 z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "4"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox["1", "2"], "-", FractionBox["\[Nu]", "2"]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", FractionBox["1", "2"]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["-", FractionBox["\[Nu]", "2"]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[TagBox[FractionBox["1", "2"], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[RowBox[List["-", FractionBox["1", SuperscriptBox["z", "2"]]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], HypergeometricPFQ] ) /; 0 < arg ( z ) π ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) Condition Proportional SphericalBesselJ ν z -1 ν -1 z 1 2 z 3 2 -1 ν ν 1 2 z -1 z ν 2 -1 HypergeometricPFQ 1 2 -1 ν 2 -1 1 -1 ν 2 -1 ν 2 -1 1 ν 2 -1 3 2 3 2 -1 1 z 2 -1 z ν 2 -1 HypergeometricPFQ 1 2 -1 ν 2 -1 ν 2 -1 1 2 ν 2 -1 1 -1 ν 2 -1 1 2 -1 1 z 2 -1 Inequality 0 z Rule z [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "\[Nu]"], " ", SqrtBox[RowBox[List["-", "z"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", " ", RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]]]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["z", "+", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"]]], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", FractionBox["\[Nu]", "2"]]], ",", RowBox[List["1", "-", FractionBox["\[Nu]", "2"]]], ",", RowBox[List["1", "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["3", "2"], "+", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", FractionBox["3", "2"], "}"]], ",", RowBox[List["-", FractionBox["1", SuperscriptBox["z", "2"]]]]]], "]"]]]], RowBox[List["2", " ", "z"]]], "+", RowBox[List[RowBox[List["Sin", "[", RowBox[List["z", "+", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"]]], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["1", "2"], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List["1", "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List["-", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", FractionBox["1", "2"], "}"]], ",", RowBox[List["-", FractionBox["1", SuperscriptBox["z", "2"]]]]]], "]"]]]]]], ")"]]]], SuperscriptBox["z", RowBox[List["3", "/", "2"]]]], "/;", RowBox[List[RowBox[List["0", "<", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", "\[Pi]"]], "&&", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]]]

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

 2007-05-02