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 SphericalBesselJ

 http://functions.wolfram.com/03.21.22.0001.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["FourierCosTransform", "[", RowBox[List[RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]", ",", "t"]], "]"]], ",", "t", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["UnitStep", "[", RowBox[List["1", "-", "z"]], "]"]], FractionBox[RowBox[List[SqrtBox["2"], " ", RowBox[List["Gamma", "[", FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], "]"]]]], RowBox[List["\[Nu]", " ", RowBox[List["Gamma", "[", FractionBox["\[Nu]", "2"], "]"]]]]], RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", FractionBox[RowBox[List["1", "+", "\[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["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]]]], 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["-", "1"]]]]]]]]]]

 MathML Form

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

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["FourierCosTransform", "[", RowBox[List[RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]_", ",", "t_"]], "]"]], ",", "t_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["UnitStep", "[", RowBox[List["1", "-", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[SqrtBox["2"], " ", RowBox[List["Gamma", "[", FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], "]"]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], ",", FractionBox["1", "2"], ",", SuperscriptBox["z", "2"]]], "]"]]]], RowBox[List["\[Nu]", " ", RowBox[List["Gamma", "[", FractionBox["\[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["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]]]], ")"]], " ", 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["-", "1"]]]]]]]]]]]]

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

 2007-05-02