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 BesselJ

 http://functions.wolfram.com/03.01.21.0029.01

 Input Form

 Integrate[Sin[b + a z] BesselJ[\[Nu], a z], z] == (z (a z)^\[Nu] (a z Cos[b] Gamma[2 + \[Nu]] HypergeometricPFQ[ {3/4 + \[Nu]/2, 1 + \[Nu]/2, 5/4 + \[Nu]/2}, {3/2, 2 + \[Nu]/2, 1 + \[Nu], 3/2 + \[Nu]}, (-a^2) z^2] + (2 + \[Nu]) Gamma[1 + \[Nu]] HypergeometricPFQ[{1/4 + \[Nu]/2, 1/2 + \[Nu]/2, 3/4 + \[Nu]/2}, {1/2, 3/2 + \[Nu]/2, 1/2 + \[Nu], 1 + \[Nu]}, (-a^2) z^2] Sin[b]))/ 2^\[Nu]/((2 + \[Nu]) Gamma[1 + \[Nu]] Gamma[2 + \[Nu]])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["Sin", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]], " ", RowBox[List["BesselJ", "[", RowBox[List["\[Nu]", ",", RowBox[List["a", " ", "z"]]]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Nu]"]]], " ", "z", " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", "z"]], ")"]], "\[Nu]"], " ", RowBox[List["(", RowBox[List[RowBox[List["a", " ", "z", " ", RowBox[List["Cos", "[", "b", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", "\[Nu]"]], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["3", "4"], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List["1", "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["5", "4"], "+", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", RowBox[List["2", "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List[FractionBox["3", "2"], "+", "\[Nu]"]]]], "}"]], ",", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], " ", SuperscriptBox["z", "2"]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["2", "+", "\[Nu]"]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "4"], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["1", "2"], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["3", "4"], "+", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", RowBox[List[FractionBox["3", "2"], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], " ", SuperscriptBox["z", "2"]]]]], "]"]], " ", RowBox[List["Sin", "[", "b", "]"]]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["2", "+", "\[Nu]"]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", "\[Nu]"]], "]"]]]], ")"]]]]]]]]

 MathML Form

 sin ( b + a z ) J ν ( a z ) z 1 ( ν + 2 ) Γ ( ν + 1 ) Γ ( ν + 2 ) ( 2 - ν z ( a z ) ν ( a z cos ( b ) Γ ( ν + 2 ) 3 F 4 ( ν 2 + 3 4 , ν 2 + 1 , ν 2 + 5 4 ; 3 2 , ν 2 + 2 , ν + 1 , ν + 3 2 ; - a 2 z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox["F", FormBox["4", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", FractionBox["3", "4"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", FractionBox["5", "4"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["3", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", "2"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Nu]", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Nu]", "+", FractionBox["3", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], " ", SuperscriptBox["z", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] + ( ν + 2 ) Γ ( ν + 1 ) 3 F 4 ( ν 2 + 1 4 , ν 2 + 1 2 , ν 2 + 3 4 ; 1 2 , ν 2 + 3 2 , ν + 1 2 , ν + 1 ; - a 2 z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox["F", FormBox["4", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", FractionBox["1", "4"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", FractionBox["1", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", FractionBox["3", "4"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["1", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", FractionBox["3", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Nu]", "+", FractionBox["1", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Nu]", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], " ", SuperscriptBox["z", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] sin ( b ) ) ) z b a z BesselJ ν a z 1 ν 2 Gamma ν 1 Gamma ν 2 -1 2 -1 ν z a z ν a z b Gamma ν 2 HypergeometricPFQ ν 2 -1 3 4 ν 2 -1 1 ν 2 -1 5 4 3 2 ν 2 -1 2 ν 1 ν 3 2 -1 a 2 z 2 ν 2 Gamma ν 1 HypergeometricPFQ ν 2 -1 1 4 ν 2 -1 1 2 ν 2 -1 3 4 1 2 ν 2 -1 3 2 ν 1 2 ν 1 -1 a 2 z 2 b [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[RowBox[List["Sin", "[", RowBox[List["b_", "+", RowBox[List["a_", " ", "z_"]]]], "]"]], " ", RowBox[List["BesselJ", "[", RowBox[List["\[Nu]_", ",", RowBox[List["a_", " ", "z_"]]]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Nu]"]]], " ", "z", " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", "z"]], ")"]], "\[Nu]"], " ", RowBox[List["(", RowBox[List[RowBox[List["a", " ", "z", " ", RowBox[List["Cos", "[", "b", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", "\[Nu]"]], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["3", "4"], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List["1", "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["5", "4"], "+", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", RowBox[List["2", "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List[FractionBox["3", "2"], "+", "\[Nu]"]]]], "}"]], ",", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], " ", SuperscriptBox["z", "2"]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["2", "+", "\[Nu]"]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "4"], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["1", "2"], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["3", "4"], "+", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", RowBox[List[FractionBox["3", "2"], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], " ", SuperscriptBox["z", "2"]]]]], "]"]], " ", RowBox[List["Sin", "[", "b", "]"]]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["2", "+", "\[Nu]"]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", "\[Nu]"]], "]"]]]]]]]]]

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

 2001-10-29