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 AiryAi

 http://functions.wolfram.com/03.05.21.0058.01

 Input Form

 Integrate[BesselK[\[Nu], (2/3) (a z^r)^(3/2)] AiryAi[a z^r], z] == (-(1/(Sqrt[Pi] r))) ((2^(-(7/3) - \[Nu]) 3^(-(7/6) - \[Nu]) z Csc[Pi \[Nu]] (4^\[Nu] ((a z^r)^(3/2))^(2 \[Nu]) MeijerG[{{(1/6) (1 - 3 \[Nu]), (1/6) (4 - 3 \[Nu]), 1 - 1/(3 r) - \[Nu]/2}, {}}, {{0, 1/3}, {1/3 - \[Nu], -\[Nu], -((2 + 3 r \[Nu])/(6 r))}}, (2/3)^(2/3) a z^r, 1/3] - 9^\[Nu] MeijerG[{{1 - 1/(3 r) + \[Nu]/2, (1/6) (1 + 3 \[Nu]), (1/6) (4 + 3 \[Nu])}, {}}, {{0, 1/3}, {\[Nu], 1/3 + \[Nu], (-2 + 3 r \[Nu])/(6 r)}}, (2/3)^(2/3) a z^r, 1/3]))/ ((a z^r)^(3/2))^\[Nu])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["BesselK", "[", RowBox[List["\[Nu]", ",", RowBox[List[FractionBox["2", "3"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["3", "/", "2"]]]]]]], "]"]], RowBox[List["AiryAi", "[", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], "]"]], " ", RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", RowBox[List[SqrtBox["\[Pi]"], " ", "r"]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", FractionBox["7", "3"]]], "-", "\[Nu]"]]], " ", SuperscriptBox["3", RowBox[List[RowBox[List["-", FractionBox["7", "6"]]], "-", "\[Nu]"]]], " ", "z", " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["3", "/", "2"]]], ")"]], RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["4", "\[Nu]"], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["3", "/", "2"]]], ")"]], RowBox[List["2", " ", "\[Nu]"]]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["3", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List["4", "-", RowBox[List["3", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List["1", "-", FractionBox["1", RowBox[List["3", " ", "r"]]], "-", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "3"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "3"], "-", "\[Nu]"]], ",", RowBox[List["-", "\[Nu]"]], ",", RowBox[List["-", FractionBox[RowBox[List["2", "+", RowBox[List["3", " ", "r", " ", "\[Nu]"]]]], RowBox[List["6", " ", "r"]]]]]]], "}"]]]], "}"]], ",", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["2", "3"], ")"]], RowBox[List["2", "/", "3"]]], " ", "a", " ", SuperscriptBox["z", "r"]]], ",", FractionBox["1", "3"]]], "]"]]]], "-", RowBox[List[SuperscriptBox["9", "\[Nu]"], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "-", FractionBox["1", RowBox[List["3", " ", "r"]]], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["3", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List["4", "+", RowBox[List["3", " ", "\[Nu]"]]]], ")"]]]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "3"]]], "}"]], ",", RowBox[List["{", RowBox[List["\[Nu]", ",", RowBox[List[FractionBox["1", "3"], "+", "\[Nu]"]], ",", FractionBox[RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["3", " ", "r", " ", "\[Nu]"]]]], RowBox[List["6", " ", "r"]]]]], "}"]]]], "}"]], ",", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["2", "3"], ")"]], RowBox[List["2", "/", "3"]]], " ", "a", " ", SuperscriptBox["z", "r"]]], ",", FractionBox["1", "3"]]], "]"]]]]]], ")"]]]], ")"]]]]]]]]

 MathML Form

 K ν ( 2 3 ( a z r ) 3 / 2 ) Ai ( a z r ) z - 1 π r ( 2 - ν - 7 3 3 - ν - 7 6 z ( ( a z r ) 3 / 2 ) - ν csc ( π ν ) ( 4 ν ( ( a z r ) 3 / 2 ) 2 ν G 3 , 5 2 , 3 ( ( 2 3 ) 2 / 3 a z r , 1 3 1 6 ( 1 - 3 ν ) , 1 6 ( 4 - 3 ν ) , - ν 2 - 1 3 r + 1 0 , 1 3 , 1 3 - ν , - ν , - 3 r ν + 2 6 r ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["3", ",", "5"]], RowBox[List["2", ",", "3"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List[TagBox[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["2", "3"], ")"]], RowBox[List["2", "/", "3"]]], " ", "a", " ", SuperscriptBox["z", "r"]]], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "3"], MeijerG, Rule[Editable, True]]]], MeijerG], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["3", " ", "\[Nu]"]]]], ")"]]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List["4", "-", RowBox[List["3", " ", "\[Nu]"]]]], ")"]]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], "-", FractionBox["1", RowBox[List["3", " ", "r"]]], "+", "1"]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "3"], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["1", "3"], "-", "\[Nu]"]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["-", "\[Nu]"]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["3", " ", "r", " ", "\[Nu]"]], "+", "2"]], RowBox[List["6", " ", "r"]]]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] - 9 ν G 3 , 5 2 , 3 ( ( 2 3 ) 2 / 3 a z r , 1 3 ν 2 - 1 3 r + 1 , 1 6 ( 3 ν + 1 ) , 1 6 ( 3 ν + 4 ) 0 , 1 3 , ν , ν + 1 3 , 3 r ν - 2 6 r ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["3", ",", "5"]], RowBox[List["2", ",", "3"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List[TagBox[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["2", "3"], ")"]], RowBox[List["2", "/", "3"]]], " ", "a", " ", SuperscriptBox["z", "r"]]], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "3"], MeijerG, Rule[Editable, True]]]], MeijerG], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "-", FractionBox["1", RowBox[List["3", " ", "r"]]], "+", "1"]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "\[Nu]"]], "+", "1"]], ")"]]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "\[Nu]"]], "+", "4"]], ")"]]]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "3"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Nu]", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Nu]", "+", FractionBox["1", "3"]]], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[RowBox[List["3", " ", "r", " ", "\[Nu]"]], "-", "2"]], RowBox[List["6", " ", "r"]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] ) ) z BesselK ν 2 3 a z r 3 2 AiryAi a z r -1 1 1 2 r -1 2 -1 ν -1 7 3 3 -1 ν -1 7 6 z a z r 3 2 -1 ν ν 4 ν a z r 3 2 2 ν MeijerG 1 6 1 -1 3 ν 1 6 4 -1 3 ν -1 ν 2 -1 -1 1 3 r -1 1 0 1 3 1 3 -1 ν -1 ν -1 3 r ν 2 6 r -1 2 3 2 3 a z r 1 3 -1 9 ν MeijerG ν 2 -1 -1 1 3 r -1 1 1 6 3 ν 1 1 6 3 ν 4 0 1 3 ν ν 1 3 3 r ν -2 6 r -1 2 3 2 3 a z r 1 3 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[RowBox[List["BesselK", "[", RowBox[List["\[Nu]_", ",", RowBox[List[FractionBox["2", "3"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a_", " ", SuperscriptBox["z_", "r_"]]], ")"]], RowBox[List["3", "/", "2"]]]]]]], "]"]], " ", RowBox[List["AiryAi", "[", RowBox[List["a_", " ", SuperscriptBox["z_", "r_"]]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", FractionBox["7", "3"]]], "-", "\[Nu]"]]], " ", SuperscriptBox["3", RowBox[List[RowBox[List["-", FractionBox["7", "6"]]], "-", "\[Nu]"]]], " ", "z", " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["3", "/", "2"]]], ")"]], RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["4", "\[Nu]"], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["3", "/", "2"]]], ")"]], RowBox[List["2", " ", "\[Nu]"]]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["3", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List["4", "-", RowBox[List["3", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List["1", "-", FractionBox["1", RowBox[List["3", " ", "r"]]], "-", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "3"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "3"], "-", "\[Nu]"]], ",", RowBox[List["-", "\[Nu]"]], ",", RowBox[List["-", FractionBox[RowBox[List["2", "+", RowBox[List["3", " ", "r", " ", "\[Nu]"]]]], RowBox[List["6", " ", "r"]]]]]]], "}"]]]], "}"]], ",", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["2", "3"], ")"]], RowBox[List["2", "/", "3"]]], " ", "a", " ", SuperscriptBox["z", "r"]]], ",", FractionBox["1", "3"]]], "]"]]]], "-", RowBox[List[SuperscriptBox["9", "\[Nu]"], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "-", FractionBox["1", RowBox[List["3", " ", "r"]]], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["3", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List["4", "+", RowBox[List["3", " ", "\[Nu]"]]]], ")"]]]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "3"]]], "}"]], ",", RowBox[List["{", RowBox[List["\[Nu]", ",", RowBox[List[FractionBox["1", "3"], "+", "\[Nu]"]], ",", FractionBox[RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["3", " ", "r", " ", "\[Nu]"]]]], RowBox[List["6", " ", "r"]]]]], "}"]]]], "}"]], ",", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["2", "3"], ")"]], RowBox[List["2", "/", "3"]]], " ", "a", " ", SuperscriptBox["z", "r"]]], ",", FractionBox["1", "3"]]], "]"]]]]]], ")"]]]], RowBox[List[SqrtBox["\[Pi]"], " ", "r"]]]]]]]]]

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

 2001-10-29