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 BesselJ

 http://functions.wolfram.com/03.01.22.0005.02

 Input Form

 HankelTransform[BesselJ[\[Nu], t], {t, \[Mu]}, z] == ((Sqrt[2] z^(1/2 + \[Mu]) Gamma[(3 + 2 \[Mu] + 2 \[Nu])/4] UnitStep[1 - z])/ (Gamma[1 + \[Mu]] Gamma[(1 - 2 \[Mu] + 2 \[Nu])/4])) Hypergeometric2F1[(3 + 2 \[Mu] - 2 \[Nu])/4, (3 + 2 \[Mu] + 2 \[Nu])/4, 1 + \[Mu], z^2] + ((Sqrt[2] z^(-1 - \[Nu]) Gamma[(3 + 2 \[Mu] + 2 \[Nu])/4] UnitStep[-1 + z])/(Gamma[1 + \[Nu]] Gamma[(1 + 2 \[Mu] - 2 \[Nu])/4])) Hypergeometric2F1[(3 - 2 \[Mu] + 2 \[Nu])/4, (3 + 2 \[Mu] + 2 \[Nu])/4, 1 + \[Nu], 1/z^2] /; z > 0 && z != 1 && Re[\[Nu] + \[Mu]] > -(3/2)

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HankelTransform", "[", RowBox[List[RowBox[List["BesselJ", "[", RowBox[List["\[Nu]", ",", "t"]], "]"]], ",", RowBox[List["{", RowBox[List["t", ",", "\[Mu]"]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SqrtBox["2"], " ", SuperscriptBox["z", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]"]]], " ", RowBox[List["Gamma", "[", FractionBox[RowBox[List["3", "+", RowBox[List["2", " ", "\[Mu]"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], "4"], "]"]], RowBox[List["UnitStep", "[", RowBox[List["1", "-", "z"]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Mu]"]], "]"]], " ", RowBox[List["Gamma", "[", FractionBox[RowBox[List["1", "-", RowBox[List["2", " ", "\[Mu]"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], "4"], "]"]]]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["3", "+", RowBox[List["2", " ", "\[Mu]"]], "-", RowBox[List["2", " ", "\[Nu]"]]]], "4"], ",", FractionBox[RowBox[List["3", "+", RowBox[List["2", " ", "\[Mu]"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], "4"], ",", RowBox[List["1", "+", "\[Mu]"]], ",", SuperscriptBox["z", "2"]]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List[SqrtBox["2"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]], " ", RowBox[List["Gamma", "[", FractionBox[RowBox[List["3", "+", RowBox[List["2", " ", "\[Mu]"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], "4"], "]"]], " ", RowBox[List["UnitStep", "[", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], "]"]]]], RowBox[List[" ", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], RowBox[List["Gamma", "[", FractionBox[RowBox[List["1", "+", RowBox[List["2", " ", "\[Mu]"]], "-", RowBox[List["2", " ", "\[Nu]"]]]], "4"], "]"]]]]]]], RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["3", "-", RowBox[List["2", " ", "\[Mu]"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], "4"], " ", ",", FractionBox[RowBox[List["3", "+", RowBox[List["2", " ", "\[Mu]"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], "4"], " ", ",", RowBox[List["1", "+", "\[Nu]"]], ",", FractionBox["1", SuperscriptBox["z", "2"]]]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List["z", ">", "0"]], "&&", RowBox[List["z", "\[NotEqual]", "1"]], "&&", RowBox[List[RowBox[List["Re", "[", RowBox[List["\[Nu]", "+", "\[Mu]"]], "]"]], ">", RowBox[List["-", FractionBox["3", "2"]]]]]]]]]]]

 MathML Form

 t ; μ [ J ν ( t ) ] ( z ) 2 z - ν - 1 Γ ( 1 4 ( 2 μ + 2 ν + 3 ) ) θ UnitStep ( z - 1 ) Γ ( ν + 1 ) Γ ( 1 4 ( 2 μ - 2 ν + 1 ) ) 2 F 1 ( 1 4 ( - 2 μ + 2 ν + 3 ) , 1 4 ( 2 μ + 2 ν + 3 ) ; ν + 1 ; 1 z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[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"]], ")"]]]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Mu]"]], "+", RowBox[List["2", " ", "\[Nu]"]], "+", "3"]], ")"]]]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["\[Nu]", "+", "1"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[FractionBox["1", SuperscriptBox["z", "2"]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] + 2 z μ + 1 2 Γ ( 1 4 ( 2 μ + 2 ν + 3 ) ) θ UnitStep ( 1 - z ) Γ ( μ + 1 ) Γ ( 1 4 ( - 2 μ + 2 ν + 1 ) ) 2 F 1 ( 1 4 ( 2 μ - 2 ν + 3 ) , 1 4 ( 2 μ + 2 ν + 3 ) ; μ + 1 ; z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[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]"]], "+", "3"]], ")"]]]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Mu]"]], "+", RowBox[List["2", " ", "\[Nu]"]], "+", "3"]], ")"]]]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["\[Mu]", "+", "1"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[SuperscriptBox["z", "2"], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] /; z > 0 z 1 Re ( μ + ν ) > - 3 2 Condition Subscript CompoundExpression t μ BesselJ ν t z 2 1 2 z -1 ν -1 Gamma 1 4 2 μ 2 ν 3 UnitStep z -1 Gamma ν 1 Gamma 1 4 2 μ -1 2 ν 1 -1 Hypergeometric2F1 1 4 -2 μ 2 ν 3 1 4 2 μ 2 ν 3 ν 1 1 z 2 -1 2 1 2 z μ 1 2 Gamma 1 4 2 μ 2 ν 3 UnitStep 1 -1 z Gamma μ 1 Gamma 1 4 -2 μ 2 ν 1 -1 Hypergeometric2F1 1 4 2 μ -1 2 ν 3 1 4 2 μ 2 ν 3 μ 1 z 2 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["BesselJ", "[", RowBox[List["\[Nu]_", ",", "t_"]], "]"]], ",", RowBox[List["{", RowBox[List["t_", ",", "\[Mu]_"]], "}"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox["2"], " ", SuperscriptBox["z", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "\[Mu]"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], "]"]], " ", RowBox[List["UnitStep", "[", RowBox[List["1", "-", "z"]], "]"]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", 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["1", "+", "\[Mu]"]], ",", SuperscriptBox["z", "2"]]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Mu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "\[Mu]"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], "]"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox["2"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "\[Mu]"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], "]"]], " ", RowBox[List["UnitStep", "[", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], "]"]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", 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["1", "+", "\[Nu]"]], ",", FractionBox["1", SuperscriptBox["z", "2"]]]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], " ", 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["\[Nu]", "+", "\[Mu]"]], "]"]], ">", RowBox[List["-", FractionBox["3", "2"]]]]]]]]]]]]]

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

 2001-10-29