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http://functions.wolfram.com/03.05.21.0054.01
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Integrate[BesselK[\[Nu], (2/3) (a z)^(3/2)] AiryAi[a z], z] ==
(-(1/(a Sqrt[Pi]))) ((2^(-3 - \[Nu]) 3^(-(1/2) - \[Nu]) Csc[Pi \[Nu]]
(4^\[Nu] ((a z)^(3/2))^(2 \[Nu])
MeijerG[{{(1 - \[Nu])/2, 1 - \[Nu]/2, 1 - \[Nu]/2}, {}},
{{1/3, 2/3}, {1/3 - \[Nu], 2/3 - \[Nu], -(\[Nu]/2)}}, (2/3)^(2/3) a z,
1/3] - 9^\[Nu] MeijerG[{{(1 + \[Nu])/2, (2 + \[Nu])/2, (2 + \[Nu])/2},
{}}, {{1/3, 2/3}, {\[Nu]/2, 1/3 + \[Nu], 2/3 + \[Nu]}},
(2/3)^(2/3) a z, 1/3]))/((a z)^(3/2))^\[Nu])
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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", " ", "z"]], ")"]], RowBox[List["3", "/", "2"]]]]]]], "]"]], RowBox[List["AiryAi", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", RowBox[List["a", " ", SqrtBox["\[Pi]"]]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "3"]], "-", "\[Nu]"]]], " ", SuperscriptBox["3", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", "z"]], ")"]], 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", " ", "z"]], ")"]], RowBox[List["3", "/", "2"]]], ")"]], RowBox[List["2", " ", "\[Nu]"]]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ",", RowBox[List["1", "-", FractionBox["\[Nu]", "2"]]], ",", RowBox[List["1", "-", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "3"], ",", FractionBox["2", "3"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "3"], "-", "\[Nu]"]], ",", RowBox[List[FractionBox["2", "3"], "-", "\[Nu]"]], ",", RowBox[List["-", FractionBox["\[Nu]", "2"]]]]], "}"]]]], "}"]], ",", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["2", "3"], ")"]], RowBox[List["2", "/", "3"]]], " ", "a", " ", "z"]], ",", FractionBox["1", "3"]]], "]"]]]], "-", RowBox[List[SuperscriptBox["9", "\[Nu]"], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], ",", FractionBox[RowBox[List["2", "+", "\[Nu]"]], "2"], ",", FractionBox[RowBox[List["2", "+", "\[Nu]"]], "2"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "3"], ",", FractionBox["2", "3"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["\[Nu]", "2"], ",", RowBox[List[FractionBox["1", "3"], "+", "\[Nu]"]], ",", RowBox[List[FractionBox["2", "3"], "+", "\[Nu]"]]]], "}"]]]], "}"]], ",", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["2", "3"], ")"]], RowBox[List["2", "/", "3"]]], " ", "a", " ", "z"]], ",", FractionBox["1", "3"]]], "]"]]]]]], ")"]]]], ")"]]]]]]]]
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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> <mrow> <msub> <mi> K </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mrow> <mfrac> <mn> 2 </mn> <mn> 3 </mn> </mfrac> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Ai </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mrow> <mi> a </mi> <mo> ⁢ </mo> <msqrt> <mi> π </mi> </msqrt> </mrow> </mfrac> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mn> 2 </mn> <mrow> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mo> - </mo> <mn> 3 </mn> </mrow> </msup> <mo> ⁢ </mo> <msup> <mn> 3 </mn> <mrow> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> csc </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mn> 4 </mn> <mi> ν </mi> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ν </mi> </mrow> </msup> <mo> ⁢ </mo> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mn> 3 </mn> <mo> , </mo> <mn> 5 </mn> </mrow> <mrow> <mn> 2 </mn> <mo> , </mo> <mn> 3 </mn> </mrow> </msubsup> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mfrac> <mn> 2 </mn> <mn> 3 </mn> </mfrac> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 3 </mn> </mfrac> </mrow> <mo> ❘ </mo> <mtable> <mtr> <mtd> <mrow> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> ν </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mi> ν </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mi> ν </mi> <mn> 2 </mn> </mfrac> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mn> 1 </mn> <mn> 3 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 2 </mn> <mn> 3 </mn> </mfrac> <mo> , </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 3 </mn> </mfrac> <mo> - </mo> <mi> ν </mi> </mrow> <mo> , </mo> <mrow> <mfrac> <mn> 2 </mn> <mn> 3 </mn> </mfrac> <mo> - </mo> <mi> ν </mi> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mi> ν </mi> <mn> 2 </mn> </mfrac> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> 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", " ", "z"]], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "3"], MeijerG, Rule[Editable, True]]]], MeijerG], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", FractionBox["\[Nu]", "2"]]], MeijerG, Rule[Editable, True]], ",", 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</mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 3 </mn> </mfrac> </mrow> <mo> ❘ </mo> <mtable> <mtr> <mtd> <mrow> <mfrac> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> , </mo> <mfrac> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> , </mo> <mfrac> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mn> 2 </mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mn> 1 </mn> <mn> 3 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 2 </mn> <mn> 3 </mn> </mfrac> <mo> , </mo> <mfrac> <mi> ν </mi> <mn> 2 </mn> </mfrac> <mo> , </mo> <mrow> <mi> ν </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 3 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mi> ν </mi> <mo> + </mo> <mfrac> <mn> 2 </mn> <mn> 3 </mn> </mfrac> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> 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type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <cn type='integer'> -3 </cn> </apply> </apply> <apply> <power /> <cn type='integer'> 3 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <apply> <csc /> <apply> <times /> <pi /> <ci> ν </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> 4 </cn> <ci> ν </ci> </apply> <apply> <power /> <apply> <power /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> ν </ci> </apply> </apply> <apply> <ci> MeijerG </ci> <list> <list> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </list> <list /> </list> <list> <list> <cn type='rational'> 1 <sep /> 3 </cn> <cn type='rational'> 2 <sep /> 3 </cn> </list> <list> <apply> <plus /> <cn type='rational'> 1 <sep /> 3 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <apply> <plus /> <cn type='rational'> 2 <sep /> 3 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </list> </list> <apply> <times /> <apply> <power /> <cn type='rational'> 2 <sep /> 3 </cn> <cn type='rational'> 2 <sep /> 3 </cn> </apply> <ci> a </ci> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <cn type='integer'> 9 </cn> <ci> ν </ci> </apply> <apply> <ci> MeijerG </ci> <list> <list> <apply> <times /> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn 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</ci> </apply> <cn type='rational'> 1 <sep /> 3 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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Date Added to functions.wolfram.com (modification date)
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