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 BesselJ

 http://functions.wolfram.com/03.01.22.0002.02

 Input Form

 FourierSinTransform[BesselJ[\[Nu], t], t, z] == UnitStep[1 - z] Sqrt[2/Pi] (Sin[\[Nu] ArcSin[z]]/Sqrt[1 - z^2]) + UnitStep[z - 1] ((2^(1/2 - \[Nu]) z^(-1 - \[Nu]))/Sqrt[Pi]) Cos[(Pi \[Nu])/2] Hypergeometric2F1[(1 + \[Nu])/2, (2 + \[Nu])/2, 1 + \[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["BesselJ", "[", RowBox[List["\[Nu]", ",", "t"]], "]"]], ",", "t", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["UnitStep", "[", RowBox[List["1", "-", "z"]], "]"]], SqrtBox[FractionBox["2", "\[Pi]"]], " ", FractionBox[RowBox[List["Sin", "[", RowBox[List["\[Nu]", " ", RowBox[List["ArcSin", "[", "z", "]"]]]], "]"]], SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]]]], "+", RowBox[List[RowBox[List["UnitStep", "[", RowBox[List["z", "-", "1"]], "]"]], FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]], " "]], SqrtBox["\[Pi]"]], RowBox[List["Cos", "[", 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["1", "+", "\[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 ) 2 1 2 - ν z - ν - 1 θ UnitStep ( z - 1 ) π cos ( π ν 2 ) 2 F 1 ( ν + 1 2 , ν + 2 2 ; 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[FractionBox[RowBox[List["\[Nu]", "+", "1"]], "2"], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["\[Nu]", "+", "2"]], "2"], 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] + θ UnitStep ( 1 - z ) 1 - z 2 2 π sin ( ν sin - 1 ( z ) ) /; z > 0 z 1 Re ( ν ) > - 2 Condition Subscript ℱ𝓈 t BesselJ ν t z 2 1 2 -1 ν z -1 ν -1 UnitStep z -1 1 2 -1 ν 2 -1 Hypergeometric2F1 ν 1 2 -1 ν 2 2 -1 ν 1 1 z 2 -1 UnitStep 1 -1 z 1 -1 z 2 1 2 -1 2 -1 1 2 ν z z 0 z 1 ν -2 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["FourierSinTransform", "[", RowBox[List[RowBox[List["BesselJ", "[", RowBox[List["\[Nu]_", ",", "t_"]], "]"]], ",", "t_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["UnitStep", "[", RowBox[List["1", "-", "z"]], "]"]], " ", SqrtBox[FractionBox["2", "\[Pi]"]], " ", RowBox[List["Sin", "[", RowBox[List["\[Nu]", " ", RowBox[List["ArcSin", "[", "z", "]"]]]], "]"]]]], SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]], "+", FractionBox[RowBox[List[RowBox[List["UnitStep", "[", RowBox[List["z", "-", "1"]], "]"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["2", 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["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], ",", FractionBox[RowBox[List["2", "+", "\[Nu]"]], "2"], ",", RowBox[List["1", "+", "\[Nu]"]], ",", FractionBox["1", SuperscriptBox["z", "2"]]]], "]"]]]], SqrtBox["\[Pi]"]]]], "/;", 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)

 2001-10-29