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http://functions.wolfram.com/03.05.21.0047.01
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Integrate[z^(\[Alpha] - 1) AiryAi[(-a) z^r] AiryAi[a z^r], z] ==
(z^\[Alpha] MeijerG[{{1 - \[Alpha]/(6 r)}, {1/6}},
{{0, 1/6, 1/3, 2/3}, {1/6, -(\[Alpha]/(6 r))}},
(a z^r)/(2^(1/3) 3^(2/3)), 1/6])/(12 2^(2/3) 3^(1/3) Pi^(3/2) r)
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "1"]]], " ", RowBox[List["AiryAi", "[", RowBox[List[RowBox[List["-", "a"]], " ", SuperscriptBox["z", "r"]]], "]"]], " ", RowBox[List["AiryAi", "[", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[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["6", " ", "r"]]]]], "}"]], ",", RowBox[List["{", FractionBox["1", "6"], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "6"], ",", FractionBox["1", "3"], ",", FractionBox["2", "3"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "6"], ",", RowBox[List["-", FractionBox["\[Alpha]", RowBox[List["6", " ", "r"]]]]]]], "}"]]]], "}"]], ",", FractionBox[RowBox[List["a", " ", SuperscriptBox["z", "r"]]], RowBox[List[SuperscriptBox["2", RowBox[List["1", "/", "3"]]], " ", SuperscriptBox["3", RowBox[List["2", "/", "3"]]]]]], ",", FractionBox["1", "6"]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List["12", " ", SuperscriptBox["2", RowBox[List["2", "/", "3"]]], " ", SuperscriptBox["3", RowBox[List["1", "/", "3"]]], " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]], " ", "r"]], ")"]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mo> ∫ </mo> <mrow> <mrow> <msup> <mi> z </mi> <mrow> <mi> α </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> Ai </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> r </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Ai </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> r </mi> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <msup> <mi> z </mi> <mi> α </mi> </msup> <mtext> </mtext> </mrow> <mrow> <mn> 12 </mn> <mo> ⁢ </mo> <msup> <mn> 2 </mn> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> <mo> ⁢ </mo> <mroot> <mn> 3 </mn> <mn> 3 </mn> </mroot> <mo> ⁢ </mo> <msup> <mi> π </mi> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> ⁢ </mo> <mi> r </mi> </mrow> </mfrac> <mo> ⁢ </mo> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mn> 2 </mn> <mo> , </mo> <mn> 6 </mn> </mrow> <mrow> <mn> 4 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> </msubsup> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <mi> a </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> r </mi> </msup> </mrow> <mrow> <mroot> <mn> 2 </mn> <mn> 3 </mn> </mroot> <mo> ⁢ </mo> <msup> <mn> 3 </mn> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> </mrow> </mfrac> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 6 </mn> </mfrac> </mrow> <mo> ❘ </mo> <mtable> <mtr> <mtd> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mi> α </mi> <mrow> <mn> 6 </mn> <mo> ⁢ </mo> <mi> r </mi> </mrow> </mfrac> </mrow> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 6 </mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn> 0 </mn> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 6 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 3 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 2 </mn> <mn> 3 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 6 </mn> </mfrac> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mi> α </mi> <mrow> <mn> 6 </mn> <mo> ⁢ </mo> <mi> r </mi> </mrow> </mfrac> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["2", ",", "6"]], RowBox[List["4", ",", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["a", " ", SuperscriptBox["z", "r"]]], RowBox[List[RadicalBox["2", "3"], " ", SuperscriptBox["3", RowBox[List["2", "/", "3"]]]]]], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "6"], MeijerG, Rule[Editable, True]]]], MeijerG], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[RowBox[List["1", "-", FractionBox["\[Alpha]", RowBox[List["6", " ", "r"]]]]], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "6"], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "6"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "3"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["2", "3"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "6"], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["-", FractionBox["\[Alpha]", RowBox[List["6", " ", "r"]]]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> α </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> AiryAi </ci> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <apply> <power /> <ci> z </ci> <ci> r </ci> </apply> </apply> </apply> <apply> <ci> AiryAi </ci> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <ci> r </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <ci> z </ci> <ci> α </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 12 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 2 <sep /> 3 </cn> </apply> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <apply> <power /> <pi /> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <ci> r </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> MeijerG </ci> <list> <list> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> α </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 6 </cn> <ci> r </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </list> <list> <cn type='rational'> 1 <sep /> 6 </cn> </list> </list> <list> <list> <cn type='integer'> 0 </cn> <cn type='rational'> 1 <sep /> 6 </cn> <cn type='rational'> 1 <sep /> 3 </cn> <cn type='rational'> 2 <sep /> 3 </cn> </list> <list> <cn type='rational'> 1 <sep /> 6 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> α </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 6 </cn> <ci> r </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </list> </list> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <ci> r </ci> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 2 <sep /> 3 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 6 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", RowBox[List["AiryAi", "[", RowBox[List[RowBox[List["-", "a_"]], " ", SuperscriptBox["z_", "r_"]]], "]"]], " ", RowBox[List["AiryAi", "[", RowBox[List["a_", " ", SuperscriptBox["z_", "r_"]]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", "-", FractionBox["\[Alpha]", RowBox[List["6", " ", "r"]]]]], "}"]], ",", RowBox[List["{", FractionBox["1", "6"], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "6"], ",", FractionBox["1", "3"], ",", FractionBox["2", "3"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "6"], ",", RowBox[List["-", FractionBox["\[Alpha]", RowBox[List["6", " ", "r"]]]]]]], "}"]]]], "}"]], ",", FractionBox[RowBox[List["a", " ", SuperscriptBox["z", "r"]]], RowBox[List[SuperscriptBox["2", RowBox[List["1", "/", "3"]]], " ", SuperscriptBox["3", RowBox[List["2", "/", "3"]]]]]], ",", FractionBox["1", "6"]]], "]"]]]], RowBox[List["12", " ", SuperscriptBox["2", RowBox[List["2", "/", "3"]]], " ", SuperscriptBox["3", RowBox[List["1", "/", "3"]]], " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]], " ", "r"]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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