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 AiryAiPrime

 http://functions.wolfram.com/03.07.21.0067.01

 Input Form

 Integrate[z^(\[Alpha] - 1) BesselK[0, (2/3) (a z^r)^(3/2)] AiryAiPrime[a z^r], z] == (-(1/(2 2^(2/3) Sqrt[Pi]))) (3^(1/6) ((1/(3 r)) (z^\[Alpha] MeijerG[{{1 - \[Alpha]/(3 r)}, {1/3, 5/6}}, {{0, 0, 2/3, 2/3}, {-(\[Alpha]/(3 r))}}, (2/3)^(2/3) a z^r, 1/3]) + (1/(3 Pi r)) (z^\[Alpha] ((3/2) Log[a z^r] - Log[(a z^r)^(3/2)]) MeijerG[{{1/3, 5/6, 1 - \[Alpha]/(3 r)}, {}}, {{0, 2/3}, {0, 2/3, -(\[Alpha]/(3 r))}}, (2/3)^(2/3) a z^r, 1/3])))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "1"]]], " ", RowBox[List["BesselK", "[", RowBox[List["0", ",", RowBox[List[FractionBox["2", "3"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["3", "/", "2"]]]]]]], "]"]], RowBox[List["AiryAiPrime", "[", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], "]"]], " ", RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", RowBox[List["2", " ", SuperscriptBox["2", RowBox[List["2", "/", "3"]]], " ", SqrtBox["\[Pi]"]]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["3", RowBox[List["1", "/", "6"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", RowBox[List["3", " ", "r"]]], RowBox[List["(", RowBox[List[SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", "-", FractionBox["\[Alpha]", RowBox[List["3", " ", "r"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "3"], ",", FractionBox["5", "6"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", "0", ",", FractionBox["2", "3"], ",", FractionBox["2", "3"]]], "}"]], ",", RowBox[List["{", RowBox[List["-", FractionBox["\[Alpha]", RowBox[List["3", " ", "r"]]]]], "}"]]]], "}"]], ",", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["2", "3"], ")"]], RowBox[List["2", "/", "3"]]], " ", "a", " ", SuperscriptBox["z", "r"]]], ",", FractionBox["1", "3"]]], "]"]]]], ")"]]]], "+", RowBox[List[FractionBox["1", RowBox[List["3", " ", "\[Pi]", " ", "r"]]], RowBox[List["(", RowBox[List[SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["3", "2"], " ", RowBox[List["Log", "[", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], "]"]]]], "-", RowBox[List["Log", "[", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["3", "/", "2"]]], "]"]]]], ")"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "3"], ",", FractionBox["5", "6"], ",", RowBox[List["1", "-", FractionBox["\[Alpha]", RowBox[List["3", " ", "r"]]]]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["2", "3"]]], "}"]], ",", RowBox[List["{", RowBox[List["0", ",", FractionBox["2", "3"], ",", RowBox[List["-", FractionBox["\[Alpha]", RowBox[List["3", " ", "r"]]]]]]], "}"]]]], "}"]], ",", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["2", "3"], ")"]], RowBox[List["2", "/", "3"]]], " ", "a", " ", SuperscriptBox["z", "r"]]], ",", FractionBox["1", "3"]]], "]"]]]], ")"]]]]]], ")"]]]], ")"]]]]]]]]

 MathML Form

 z α - 1 K 0 ( 2 3 ( a z r ) 3 / 2 ) Ai ( a z r ) z - 3 6 2 2 2 / 3 π ( z α 3 r G 3 , 5 4 , 1 ( ( 2 3 ) 2 / 3 a z r , 1 3 1 - α 3 r , 1 3 , 5 6 0 , 0 , 2 3 , 2 3 , - α 3 r ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["3", ",", "5"]], RowBox[List["4", ",", "1"]]], "\[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["1", "-", FractionBox["\[Alpha]", RowBox[List["3", " ", "r"]]]]], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "3"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["5", "6"], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["2", "3"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["2", "3"], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["-", FractionBox["\[Alpha]", RowBox[List["3", " ", "r"]]]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] + z α ( 3 2 log ( a z r ) - log ( ( a z r ) 3 / 2 ) ) 3 π r G 3 , 5 2 , 3 ( ( 2 3 ) 2 / 3 a z r , 1 3 1 3 , 5 6 , 1 - α 3 r 0 , 2 3 , 0 , 2 3 , - α 3 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[FractionBox["1", "3"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["5", "6"], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", FractionBox["\[Alpha]", RowBox[List["3", " ", "r"]]]]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["2", "3"], MeijerG, Rule[Editable, True]], ",", TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["2", "3"], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["-", FractionBox["\[Alpha]", RowBox[List["3", " ", "r"]]]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] ) z z α -1 BesselK 0 2 3 a z r 3 2 AiryAiPrime a z r -1 3 1 6 2 2 2 3 1 2 -1 z α 3 r -1 MeijerG 1 -1 α 3 r -1 1 3 5 6 0 0 2 3 2 3 -1 α 3 r -1 2 3 2 3 a z r 1 3 z α 3 2 a z r -1 a z r 3 2 3 r -1 MeijerG 1 3 5 6 1 -1 α 3 r -1 0 2 3 0 2 3 -1 α 3 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[SuperscriptBox["z_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", RowBox[List["BesselK", "[", RowBox[List["0", ",", RowBox[List[FractionBox["2", "3"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a_", " ", SuperscriptBox["z_", "r_"]]], ")"]], RowBox[List["3", "/", "2"]]]]]]], "]"]], " ", RowBox[List["AiryAiPrime", "[", RowBox[List["a_", " ", SuperscriptBox["z_", "r_"]]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["3", RowBox[List["1", "/", "6"]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", "-", FractionBox["\[Alpha]", RowBox[List["3", " ", "r"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "3"], ",", FractionBox["5", "6"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", "0", ",", FractionBox["2", "3"], ",", FractionBox["2", "3"]]], "}"]], ",", RowBox[List["{", RowBox[List["-", FractionBox["\[Alpha]", RowBox[List["3", " ", "r"]]]]], "}"]]]], "}"]], ",", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["2", "3"], ")"]], RowBox[List["2", "/", "3"]]], " ", "a", " ", SuperscriptBox["z", "r"]]], ",", FractionBox["1", "3"]]], "]"]]]], RowBox[List["3", " ", "r"]]], "+", FractionBox[RowBox[List[SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["3", "2"], " ", RowBox[List["Log", "[", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], "]"]]]], "-", RowBox[List["Log", "[", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["3", "/", "2"]]], "]"]]]], ")"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "3"], ",", FractionBox["5", "6"], ",", RowBox[List["1", "-", FractionBox["\[Alpha]", RowBox[List["3", " ", "r"]]]]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["2", "3"]]], "}"]], ",", RowBox[List["{", RowBox[List["0", ",", FractionBox["2", "3"], ",", RowBox[List["-", FractionBox["\[Alpha]", RowBox[List["3", " ", "r"]]]]]]], "}"]]]], "}"]], ",", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["2", "3"], ")"]], RowBox[List["2", "/", "3"]]], " ", "a", " ", SuperscriptBox["z", "r"]]], ",", FractionBox["1", "3"]]], "]"]]]], RowBox[List["3", " ", "\[Pi]", " ", "r"]]]]], ")"]]]], RowBox[List["2", " ", SuperscriptBox["2", RowBox[List["2", "/", "3"]]], " ", SqrtBox["\[Pi]"]]]]]]]]]]

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

 2001-10-29