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 SphericalBesselJ

 http://functions.wolfram.com/03.21.22.0008.01

 Input Form

 HankelTransform[SphericalBesselJ[\[Nu], t], {t, \[Mu]}, z] == UnitStep[1 - z] (((-1)^(\[Mu]/4) Sqrt[Pi/2] Sqrt[z] ((-(-1)^(3/4)) z)^\[Mu] Gamma[(1/4) (3 + 2 \[Mu] + 2 \[Nu])])/ Gamma[(1/4) (3 - 2 \[Mu] + 2 \[Nu])]) Hypergeometric2F1Regularized[ (1/4) (1 + 2 \[Mu] - 2 \[Nu]), (1/4) (3 + 2 \[Mu] + 2 \[Nu]), 1 + \[Mu], z^2] + UnitStep[z - 1] ((Sqrt[Pi/2] z^(-(3/2) - \[Nu]) Gamma[(1/4) (3 + 2 \[Mu] + 2 \[Nu])])/ Gamma[(1/4) (1 + 2 \[Mu] - 2 \[Nu])]) Hypergeometric2F1Regularized[ (1/4) (3 - 2 \[Mu] + 2 \[Nu]), (1/4) (3 + 2 \[Mu] + 2 \[Nu]), 3/2 + \[Nu], 1/z^2] /; z > 0 && z != 1 && Re[\[Mu] + \[Nu]] > -(3/2)

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HankelTransform", "[", RowBox[List[RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]", ",", "t"]], "]"]], ",", RowBox[List["{", RowBox[List["t", ",", "\[Mu]"]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["UnitStep", "[", RowBox[List["1", "-", "z"]], "]"]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["\[Mu]", "/", "4"]]], " ", SqrtBox[FractionBox["\[Pi]", "2"]], " ", SqrtBox["z"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]]]], " ", "z"]], ")"]], "\[Mu]"], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "\[Mu]"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], "]"]], " "]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "-", RowBox[List["2", " ", "\[Mu]"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], "]"]]], RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Mu]"]], "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "\[Mu]"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List["1", "+", "\[Mu]"]], ",", SuperscriptBox["z", "2"]]], "]"]]]], "+", RowBox[List[RowBox[List["UnitStep", "[", RowBox[List["z", "-", "1"]], "]"]], FractionBox[RowBox[List[SqrtBox[FractionBox["\[Pi]", "2"]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", FractionBox["3", "2"]]], "-", "\[Nu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "\[Mu]"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Mu]"]], "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], "]"]]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "-", RowBox[List["2", " ", "\[Mu]"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "\[Mu]"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List[FractionBox["3", "2"], "+", "\[Nu]"]], ",", FractionBox["1", SuperscriptBox["z", "2"]]]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List["z", ">", "0"]], "&&", RowBox[List["z", "\[NotEqual]", "1"]], "&&", RowBox[List[RowBox[List["Re", "[", RowBox[List["\[Mu]", "+", "\[Nu]"]], "]"]], ">", RowBox[List["-", FractionBox["3", "2"]]]]]]]]]]]

 MathML Form

 t ; μ [ j ν ( t ) ] ( z ) θ UnitStep ( 1 - z ) Γ ( 1 4 ( - 2 μ + 2 ν + 3 ) ) ( - 1 ) μ / 4 π 2 z ( - ( - 1 ) 3 / 4 z ) μ Γ ( 1 4 ( 2 μ + 2 ν + 3 ) ) 2 F ~ 1 ( 1 4 ( 2 μ - 2 ν + 1 ) , 1 4 ( 2 μ + 2 ν + 3 ) ; μ + 1 ; z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox[OverscriptBox["F", "~"], "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Mu]"]], "-", RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], ")"]]]], Hypergeometric2F1Regularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Mu]"]], "+", RowBox[List["2", " ", "\[Nu]"]], "+", "3"]], ")"]]]], Hypergeometric2F1Regularized, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["\[Mu]", "+", "1"]], Hypergeometric2F1Regularized, Rule[Editable, True], Rule[Selectable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[SuperscriptBox["z", "2"], Hypergeometric2F1Regularized, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], Hypergeometric2F1Regularized] + θ UnitStep ( z - 1 ) Γ ( 1 4 ( 2 μ - 2 ν + 1 ) ) π 2 z - ν - 3 2 Γ ( 1 4 ( 2 μ + 2 ν + 3 ) ) 2 F ~ 1 ( 1 4 ( - 2 μ + 2 ν + 3 ) , 1 4 ( 2 μ + 2 ν + 3 ) ; ν + 3 2 ; 1 z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox[OverscriptBox["F", "~"], "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[Mu]"]], "+", RowBox[List["2", " ", "\[Nu]"]], "+", "3"]], ")"]]]], Hypergeometric2F1Regularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Mu]"]], "+", RowBox[List["2", " ", "\[Nu]"]], "+", "3"]], ")"]]]], Hypergeometric2F1Regularized, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["\[Nu]", "+", FractionBox["3", "2"]]], Hypergeometric2F1Regularized, Rule[Editable, True], Rule[Selectable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[FractionBox["1", SuperscriptBox["z", "2"]], Hypergeometric2F1Regularized, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], Hypergeometric2F1Regularized] /; z > 0 z 1 Re ( μ + ν ) > - 3 2 Condition Subscript CompoundExpression t μ SphericalBesselJ ν t z UnitStep 1 -1 z Gamma 1 4 -2 μ 2 ν 3 -1 -1 μ 4 -1 2 -1 1 2 z 1 2 -1 -1 3 4 z μ Gamma 1 4 2 μ 2 ν 3 Hypergeometric2F1Regularized 1 4 2 μ -1 2 ν 1 1 4 2 μ 2 ν 3 μ 1 z 2 UnitStep z -1 Gamma 1 4 2 μ -1 2 ν 1 -1 2 -1 1 2 z -1 ν -1 3 2 Gamma 1 4 2 μ 2 ν 3 Hypergeometric2F1Regularized 1 4 -2 μ 2 ν 3 1 4 2 μ 2 ν 3 ν 3 2 1 z 2 -1 z 0 z 1 μ ν -1 3 2 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HankelTransform", "[", RowBox[List[RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]_", ",", "t_"]], "]"]], ",", RowBox[List["{", RowBox[List["t_", ",", "\[Mu]_"]], "}"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["UnitStep", "[", RowBox[List["1", "-", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["\[Mu]", "/", "4"]]], " ", SqrtBox[FractionBox["\[Pi]", "2"]], " ", SqrtBox["z"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]]]], " ", "z"]], ")"]], "\[Mu]"], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "\[Mu]"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], "]"]]]], ")"]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Mu]"]], "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "\[Mu]"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List["1", "+", "\[Mu]"]], ",", SuperscriptBox["z", "2"]]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "-", RowBox[List["2", " ", "\[Mu]"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], "]"]]], "+", FractionBox[RowBox[List[RowBox[List["UnitStep", "[", RowBox[List["z", "-", "1"]], "]"]], " ", RowBox[List["(", RowBox[List[SqrtBox[FractionBox["\[Pi]", "2"]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", FractionBox["3", "2"]]], "-", "\[Nu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "\[Mu]"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], "]"]]]], ")"]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "-", RowBox[List["2", " ", "\[Mu]"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "\[Mu]"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List[FractionBox["3", "2"], "+", "\[Nu]"]], ",", FractionBox["1", SuperscriptBox["z", "2"]]]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Mu]"]], "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], "]"]]]]], "/;", RowBox[List[RowBox[List["z", ">", "0"]], "&&", RowBox[List["z", "\[NotEqual]", "1"]], "&&", RowBox[List[RowBox[List["Re", "[", RowBox[List["\[Mu]", "+", "\[Nu]"]], "]"]], ">", RowBox[List["-", FractionBox["3", "2"]]]]]]]]]]]]]

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

 2007-05-02