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http://functions.wolfram.com/03.05.21.0058.01
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Integrate[BesselK[\[Nu], (2/3) (a z^r)^(3/2)] AiryAi[a z^r], z] ==
(-(1/(Sqrt[Pi] r))) ((2^(-(7/3) - \[Nu]) 3^(-(7/6) - \[Nu]) z Csc[Pi \[Nu]]
(4^\[Nu] ((a z^r)^(3/2))^(2 \[Nu])
MeijerG[{{(1/6) (1 - 3 \[Nu]), (1/6) (4 - 3 \[Nu]),
1 - 1/(3 r) - \[Nu]/2}, {}}, {{0, 1/3}, {1/3 - \[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) (1 + 3 \[Nu]),
(1/6) (4 + 3 \[Nu])}, {}}, {{0, 1/3}, {\[Nu], 1/3 + \[Nu],
(-2 + 3 r \[Nu])/(6 r)}}, (2/3)^(2/3) a z^r, 1/3]))/
((a z^r)^(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", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["3", "/", "2"]]]]]]], "]"]], RowBox[List["AiryAi", "[", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], "]"]], " ", RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", RowBox[List[SqrtBox["\[Pi]"], " ", "r"]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", FractionBox["7", "3"]]], "-", "\[Nu]"]]], " ", SuperscriptBox["3", RowBox[List[RowBox[List["-", FractionBox["7", "6"]]], "-", "\[Nu]"]]], " ", "z", " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], 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", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["3", "/", "2"]]], ")"]], RowBox[List["2", " ", "\[Nu]"]]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["3", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List["4", "-", RowBox[List["3", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List["1", "-", FractionBox["1", RowBox[List["3", " ", "r"]]], "-", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "3"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "3"], "-", "\[Nu]"]], ",", RowBox[List["-", "\[Nu]"]], ",", RowBox[List["-", FractionBox[RowBox[List["2", "+", RowBox[List["3", " ", "r", " ", "\[Nu]"]]]], RowBox[List["6", " ", "r"]]]]]]], "}"]]]], "}"]], ",", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["2", "3"], ")"]], RowBox[List["2", "/", "3"]]], " ", "a", " ", SuperscriptBox["z", "r"]]], ",", FractionBox["1", "3"]]], "]"]]]], "-", RowBox[List[SuperscriptBox["9", "\[Nu]"], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "-", FractionBox["1", RowBox[List["3", " ", "r"]]], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["3", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List["4", "+", RowBox[List["3", " ", "\[Nu]"]]]], ")"]]]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "3"]]], "}"]], ",", RowBox[List["{", RowBox[List["\[Nu]", ",", RowBox[List[FractionBox["1", "3"], "+", "\[Nu]"]], ",", FractionBox[RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["3", " ", "r", " ", "\[Nu]"]]]], RowBox[List["6", " ", "r"]]]]], "}"]]]], "}"]], ",", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["2", "3"], ")"]], RowBox[List["2", "/", "3"]]], " ", "a", " ", SuperscriptBox["z", "r"]]], ",", 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> <msup> <mi> z </mi> <mi> r </mi> </msup> </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> <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> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mrow> <msqrt> <mi> π </mi> </msqrt> <mo> ⁢ </mo> <mi> r </mi> </mrow> </mfrac> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mn> 2 </mn> <mrow> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mo> - </mo> <mfrac> <mn> 7 </mn> <mn> 3 </mn> </mfrac> </mrow> </msup> <mo> ⁢ </mo> <msup> <mn> 3 </mn> <mrow> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mo> - </mo> <mfrac> <mn> 7 </mn> <mn> 6 </mn> </mfrac> </mrow> </msup> <mo> ⁢ </mo> <mi> z </mi> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> r </mi> </msup> </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> <msup> <mi> z </mi> <mi> r </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> 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<list> <list> <apply> <times /> <cn type='rational'> 1 <sep /> 6 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <ci> ν </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 6 </cn> <apply> <plus /> <cn type='integer'> 4 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <ci> ν </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <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> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> r </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </list> <list /> </list> <list> <list> <cn type='integer'> 0 </cn> <cn type='rational'> 1 <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> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> r </ci> <ci> ν </ci> </apply> <cn type='integer'> 2 </cn> </apply> <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 /> <apply> <power /> <cn type='rational'> 2 <sep /> 3 </cn> <cn type='rational'> 2 <sep /> 3 </cn> </apply> <ci> a </ci> <apply> <power /> <ci> z </ci> <ci> r </ci> </apply> </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> <plus /> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> r </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 6 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> ν </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 6 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> ν </ci> </apply> <cn type='integer'> 4 </cn> </apply> </apply> </list> <list /> </list> <list> <list> <cn type='integer'> 0 </cn> <cn type='rational'> 1 <sep /> 3 </cn> </list> <list> <ci> ν </ci> <apply> <plus /> <ci> ν </ci> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> r </ci> <ci> ν </ci> </apply> <cn type='integer'> -2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 6 </cn> <ci> r </ci> </apply> <cn type='integer'> -1 </cn> </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> <apply> <power /> <ci> z </ci> <ci> r </ci> </apply> </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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