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 AiryBiPrime

 http://functions.wolfram.com/03.08.21.0056.01

 Input Form

 Integrate[BesselK[\[Nu], (2/3) (a z^r)^(3/2)] AiryBiPrime[a z^r], z] == (-(1/r)) ((2^(-(5/3) - \[Nu]) 3^(-(5/6) - \[Nu]) Sqrt[Pi] z Csc[Pi \[Nu]] (4^\[Nu] ((a z^r)^(3/2))^(2 \[Nu]) MeijerG[{{(1/6) (2 - 3 \[Nu]), (1/6) (5 - 3 \[Nu]), 1 - 1/(3 r) - \[Nu]/2}, {1/3}}, {{0, 2/3}, {1/3, 2/3 - \[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) (2 + 3 \[Nu]), (1/6) (5 + 3 \[Nu])}, {1/3}}, {{0, 2/3}, {1/3, \[Nu], 2/3 + \[Nu], (-2 + 3 r \[Nu])/(6 r)}}, (-(2/3)^(2/3)) a z^r, 1/3]))/ ((a z^r)^(3/2))^\[Nu])

 Standard Form

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 MathML Form

 K ν ( 2 3 ( a z r ) 3 / 2 ) Bi ( a z r ) z - 1 r ( 2 - ν - 5 3 3 - ν - 5 6 π z ( ( a z r ) 3 / 2 ) - ν csc ( π ν ) ( 4 ν ( ( a z r ) 3 / 2 ) 2 ν G 4 , 6 2 , 3 ( - ( 2 3 ) 2 / 3 a z r , 1 3 1 6 ( 2 - 3 ν ) , 1 6 ( 5 - 3 ν ) , - ν 2 - 1 3 r + 1 , 1 3 0 , 2 3 , 1 3 , 2 3 - ν , - ν , - 3 r ν + 2 6 r ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["4", ",", "6"]], RowBox[List["2", ",", "3"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List[TagBox[RowBox[List[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["2", "-", RowBox[List["3", " ", "\[Nu]"]]]], ")"]]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List["5", "-", 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]], ",", TagBox[FractionBox["1", "3"], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["2", "3"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "3"], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["2", "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 4 , 6 2 , 3 ( - ( 2 3 ) 2 / 3 a z r , 1 3 ν 2 - 1 3 r + 1 , 1 6 ( 3 ν + 2 ) , 1 6 ( 3 ν + 5 ) , 1 3 0 , 2 3 , 1 3 , ν , ν + 2 3 , 3 r ν - 2 6 r ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["4", ",", "6"]], RowBox[List["2", ",", "3"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List[TagBox[RowBox[List[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]"]], "+", "2"]], ")"]]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "\[Nu]"]], "+", "5"]], ")"]]]], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "3"], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["2", "3"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "3"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Nu]", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Nu]", "+", FractionBox["2", "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 AiryBiPrime a z r -1 1 r -1 2 -1 ν -1 5 3 3 -1 ν -1 5 6 1 2 z a z r 3 2 -1 ν ν 4 ν a z r 3 2 2 ν MeijerG 1 6 2 -1 3 ν 1 6 5 -1 3 ν -1 ν 2 -1 -1 1 3 r -1 1 1 3 0 2 3 1 3 2 3 -1 ν -1 ν -1 3 r ν 2 6 r -1 -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 2 1 6 3 ν 5 1 3 0 2 3 1 3 ν ν 2 3 3 r ν -2 6 r -1 -1 2 3 2 3 a z r 1 3 [/itex]

 Rule Form

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

 2001-10-29