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 AiryBiPrime

 http://functions.wolfram.com/03.08.21.0083.01

 Input Form

 Integrate[z^(\[Alpha] - 1) AiryBi[a z^r] AiryBiPrime[a z^r], z] == (1/(12 Pi^(3/2) r \[Alpha])) (z^\[Alpha] (6 Sqrt[3 Pi] r + Sqrt[3] \[Alpha] MeijerG[{{1/2, 1 - \[Alpha]/(3 r)}, {}}, {{1/3, 2/3}, {0, -(\[Alpha]/(3 r))}}, (2/3)^(2/3) a z^r, 1/3] + 8 Pi^2 \[Alpha] MeijerG[{{1/2, 1, 1 - \[Alpha]/(3 r)}, {5/6, 4/3}}, {{2/3, 1}, {0, 1/3, 5/6, 4/3, -(\[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["AiryBi", "[", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], "]"]], RowBox[List["AiryBiPrime", "[", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], "]"]], " ", RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["12", " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]], " ", "r", " ", "\[Alpha]"]]], RowBox[List["(", RowBox[List[SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List["(", RowBox[List[RowBox[List["6", " ", SqrtBox[RowBox[List["3", " ", "\[Pi]"]]], " ", "r"]], "+", RowBox[List[SqrtBox["3"], " ", "\[Alpha]", " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["1", "-", FractionBox["\[Alpha]", RowBox[List["3", " ", "r"]]]]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "3"], ",", FractionBox["2", "3"]]], "}"]], ",", RowBox[List["{", RowBox[List["0", ",", 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["8", " ", SuperscriptBox["\[Pi]", "2"], " ", "\[Alpha]", " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", "1", ",", RowBox[List["1", "-", FractionBox["\[Alpha]", RowBox[List["3", " ", "r"]]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["5", "6"], ",", FractionBox["4", "3"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["2", "3"], ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "3"], ",", FractionBox["5", "6"], ",", FractionBox["4", "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 Bi ( a z r ) Bi ( a z r ) z z α 12 π 3 / 2 r α ( 6 3 π r + 3 α G 2 , 4 2 , 2 ( ( 2 3 ) 2 / 3 a z r , 1 3 1 2 , 1 - α 3 r 1 3 , 2 3 , 0 , - α 3 r ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["2", ",", "4"]], RowBox[List["2", ",", "2"]]], "\[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", "2"], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", FractionBox["\[Alpha]", RowBox[List["3", " ", "r"]]]]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[FractionBox["1", "3"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["2", "3"], MeijerG, Rule[Editable, True]], ",", TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["-", FractionBox["\[Alpha]", RowBox[List["3", " ", "r"]]]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] + 8 π 2 α G 5 , 7 2 , 3 ( ( 2 3 ) 2 / 3 a z r , 1 3 1 2 , 1 , 1 - α 3 r , 5 6 , 4 3 2 3 , 1 , 0 , 1 3 , 5 6 , 4 3 , - α 3 r ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["5", ",", "7"]], 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", "2"], MeijerG, Rule[Editable, True]], ",", TagBox["1", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", FractionBox["\[Alpha]", RowBox[List["3", " ", "r"]]]]], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["5", "6"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["4", "3"], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[FractionBox["2", "3"], MeijerG, Rule[Editable, True]], ",", TagBox["1", MeijerG, Rule[Editable, True]], ",", TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "3"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["5", "6"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["4", "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 AiryBi a z r AiryBiPrime a z r z α 12 3 2 r α -1 6 3 1 2 r 3 1 2 α MeijerG 1 2 1 -1 α 3 r -1 1 3 2 3 0 -1 α 3 r -1 2 3 2 3 a z r 1 3 8 2 α MeijerG 1 2 1 1 -1 α 3 r -1 5 6 4 3 2 3 1 0 1 3 5 6 4 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["AiryBi", "[", RowBox[List["a_", " ", SuperscriptBox["z_", "r_"]]], "]"]], " ", RowBox[List["AiryBiPrime", "[", RowBox[List["a_", " ", SuperscriptBox["z_", "r_"]]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List["(", RowBox[List[RowBox[List["6", " ", SqrtBox[RowBox[List["3", " ", "\[Pi]"]]], " ", "r"]], "+", RowBox[List[SqrtBox["3"], " ", "\[Alpha]", " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["1", "-", FractionBox["\[Alpha]", RowBox[List["3", " ", "r"]]]]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "3"], ",", FractionBox["2", "3"]]], "}"]], ",", RowBox[List["{", RowBox[List["0", ",", 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["8", " ", SuperscriptBox["\[Pi]", "2"], " ", "\[Alpha]", " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", "1", ",", RowBox[List["1", "-", FractionBox["\[Alpha]", RowBox[List["3", " ", "r"]]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["5", "6"], ",", FractionBox["4", "3"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["2", "3"], ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "3"], ",", FractionBox["5", "6"], ",", FractionBox["4", "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["12", " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]], " ", "r", " ", "\[Alpha]"]]]]]]]

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

 2001-10-29