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 SphericalBesselJ

 http://functions.wolfram.com/03.21.22.0002.01

 Input Form

 FourierSinTransform[SphericalBesselJ[\[Nu], t], t, z] == (-UnitStep[1 - z]) (1/(2 Sqrt[2] z Gamma[(3 + \[Nu])/2])) Gamma[\[Nu]/2] ((-1 + z^2) Hypergeometric2F1[(1 - \[Nu])/2, (2 + \[Nu])/2, -(1/2), z^2] + (1 - 3 z^2) Hypergeometric2F1[(1 - \[Nu])/2, (2 + \[Nu])/2, 1/2, z^2]) + UnitStep[z - 1] ((2^(-(1/2) - \[Nu]) z^(-1 - \[Nu]) Cos[(Pi \[Nu])/2] Gamma[1 + \[Nu]])/Gamma[3/2 + \[Nu]]) Hypergeometric2F1[(1 + \[Nu])/2, (2 + \[Nu])/2, 3/2 + \[Nu], 1/z^2] /; z > 0 && z != 1 && Re[\[Nu]] > -2

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["FourierSinTransform", "[", RowBox[List[RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]", ",", "t"]], "]"]], ",", "t", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["UnitStep", "[", RowBox[List["1", "-", "z"]], "]"]]]], FractionBox["1", RowBox[List["2", " ", SqrtBox["2"], " ", "z", " ", RowBox[List["Gamma", "[", FractionBox[RowBox[List["3", "+", "\[Nu]"]], "2"], "]"]]]]], RowBox[List["Gamma", "[", FractionBox["\[Nu]", "2"], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ",", FractionBox[RowBox[List["2", "+", "\[Nu]"]], "2"], ",", RowBox[List["-", FractionBox["1", "2"]]], ",", SuperscriptBox["z", "2"]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", RowBox[List["3", " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ",", FractionBox[RowBox[List["2", "+", "\[Nu]"]], "2"], ",", FractionBox["1", "2"], ",", SuperscriptBox["z", "2"]]], "]"]]]]]], ")"]]]], "+", RowBox[List[RowBox[List["UnitStep", "[", RowBox[List["z", "-", "1"]], "]"]], FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "\[Nu]"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]], " ", RowBox[List["Cos", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "+", "\[Nu]"]], "]"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], ",", FractionBox[RowBox[List["2", "+", "\[Nu]"]], "2"], ",", RowBox[List[FractionBox["3", "2"], "+", "\[Nu]"]], ",", FractionBox["1", SuperscriptBox["z", "2"]]]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List["z", ">", "0"]], "\[And]", RowBox[List["z", "\[NotEqual]", "1"]], "\[And]", RowBox[List[RowBox[List["Re", "[", "\[Nu]", "]"]], ">", RowBox[List["-", "2"]]]]]]]]]]

 MathML Form

 ℱ𝓈 t [ j ν ( t ) ] ( z ) θ UnitStep ( z - 1 ) 2 - ν - 1 2 z - ν - 1 cos ( π ν 2 ) Γ ( ν + 1 ) Γ ( ν + 3 2 ) 2 F 1 ( ν + 1 2 , ν + 2 2 ; ν + 3 2 ; 1 z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["\[Nu]", "+", "1"]], "2"], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox[RowBox[List["\[Nu]", "+", "2"]], "2"], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["\[Nu]", "+", FractionBox["3", "2"]]], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[FractionBox["1", SuperscriptBox["z", "2"]], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], Hypergeometric2F1] - θ UnitStep ( 1 - z ) 2 2 z Γ ( ν + 3 2 ) Γ ( ν 2 ) ( ( 1 - 3 z 2 ) 2 F 1 ( 1 - ν 2 , ν + 2 2 ; 1 2 ; z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox[RowBox[List["\[Nu]", "+", "2"]], "2"], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[TagBox[FractionBox["1", "2"], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[SuperscriptBox["z", "2"], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], Hypergeometric2F1] + ( z 2 - 1 ) 2 F 1 ( 1 - ν 2 , ν + 2 2 ; - 1 2 ; z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox[RowBox[List["\[Nu]", "+", "2"]], "2"], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["-", FractionBox["1", "2"]]], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[SuperscriptBox["z", "2"], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], Hypergeometric2F1] ) /; z > 0 z 1 Re ( ν ) > - 2 Condition Subscript ℱ𝓈 t SphericalBesselJ ν t z UnitStep z -1 2 -1 ν -1 1 2 z -1 ν -1 ν 2 -1 Gamma ν 1 Gamma ν 3 2 -1 Hypergeometric2F1 ν 1 2 -1 ν 2 2 -1 ν 3 2 1 z 2 -1 -1 UnitStep 1 -1 z 2 2 1 2 z Gamma ν 3 2 -1 -1 Gamma ν 2 -1 1 -1 3 z 2 Hypergeometric2F1 1 -1 ν 2 -1 ν 2 2 -1 1 2 z 2 z 2 -1 Hypergeometric2F1 1 -1 ν 2 -1 ν 2 2 -1 -1 1 2 z 2 z 0 z 1 ν -2 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["FourierSinTransform", "[", RowBox[List[RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]_", ",", "t_"]], "]"]], ",", "t_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["UnitStep", "[", RowBox[List["1", "-", "z"]], "]"]], " ", RowBox[List["Gamma", "[", FractionBox["\[Nu]", "2"], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ",", FractionBox[RowBox[List["2", "+", "\[Nu]"]], "2"], ",", RowBox[List["-", FractionBox["1", "2"]]], ",", SuperscriptBox["z", "2"]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", RowBox[List["3", " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ",", FractionBox[RowBox[List["2", "+", "\[Nu]"]], "2"], ",", FractionBox["1", "2"], ",", SuperscriptBox["z", "2"]]], "]"]]]]]], ")"]]]], RowBox[List["2", " ", SqrtBox["2"], " ", "z", " ", RowBox[List["Gamma", "[", FractionBox[RowBox[List["3", "+", "\[Nu]"]], "2"], "]"]]]]]]], "+", FractionBox[RowBox[List[RowBox[List["UnitStep", "[", RowBox[List["z", "-", "1"]], "]"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "\[Nu]"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]], " ", RowBox[List["Cos", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], ",", FractionBox[RowBox[List["2", "+", "\[Nu]"]], "2"], ",", RowBox[List[FractionBox["3", "2"], "+", "\[Nu]"]], ",", FractionBox["1", SuperscriptBox["z", "2"]]]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "+", "\[Nu]"]], "]"]]]]], "/;", RowBox[List[RowBox[List["z", ">", "0"]], "&&", RowBox[List["z", "\[NotEqual]", "1"]], "&&", RowBox[List[RowBox[List["Re", "[", "\[Nu]", "]"]], ">", RowBox[List["-", "2"]]]]]]]]]]]]

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

 2007-05-02