Derivative of the Airy function Bi: Integration (formula 03.08.21.0056)

AiryBiPrime

$Mathematica Notation$

Bessel-Type Functions

AiryBiPrime[z]

Integration

Indefinite integration

Involving direct function and Bessel-type functions

Involving Bessel functions

Involving Bessel K

Power arguments

http://functions.wolfram.com/03.08.21.0056.01

Input Form

Integrate[BesselK[\[Nu], (2/3) (a z^r)^(3/2)] AiryBiPrime[a z^r], z] == (-(1/r)) ((2^(-(5/3) - \[Nu]) 3^(-(5/6) - \[Nu]) Sqrt[Pi] z Csc[Pi \[Nu]] (4^\[Nu] ((a z^r)^(3/2))^(2 \[Nu]) MeijerG[{{(1/6) (2 - 3 \[Nu]), (1/6) (5 - 3 \[Nu]), 1 - 1/(3 r) - \[Nu]/2}, {1/3}}, {{0, 2/3}, {1/3, 2/3 - \[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) (2 + 3 \[Nu]), (1/6) (5 + 3 \[Nu])}, {1/3}}, {{0, 2/3}, {1/3, \[Nu], 2/3 + \[Nu], (-2 + 3 r \[Nu])/(6 r)}}, (-(2/3)^(2/3)) a z^r, 1/3]))/ ((a z^r)^(3/2))^\[Nu])

Standard Form

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["AiryBiPrime", "[", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], "]"]], " ", RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", "r"]]], RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", FractionBox["5", "3"]]], "-", "\[Nu]"]]], " ", SuperscriptBox["3", RowBox[List[RowBox[List["-", FractionBox["5", "6"]]], "-", "\[Nu]"]]], " ", SqrtBox["\[Pi]"], " ", "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["2", "-", RowBox[List["3", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List["5", "-", RowBox[List["3", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List["1", "-", FractionBox["1", RowBox[List["3", " ", "r"]]], "-", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", FractionBox["1", "3"], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["2", "3"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "3"], ",", RowBox[List[FractionBox["2", "3"], "-", "\[Nu]"]], ",", RowBox[List["-", "\[Nu]"]], ",", RowBox[List["-", FractionBox[RowBox[List["2", "+", RowBox[List["3", " ", "r", " ", "\[Nu]"]]]], RowBox[List["6", " ", "r"]]]]]]], "}"]]]], "}"]], ",", RowBox[List[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["2", "+", RowBox[List["3", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List["5", "+", RowBox[List["3", " ", "\[Nu]"]]]], ")"]]]]]], "}"]], ",", RowBox[List["{", FractionBox["1", "3"], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["2", "3"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "3"], ",", "\[Nu]", ",", RowBox[List[FractionBox["2", "3"], "+", "\[Nu]"]], ",", FractionBox[RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["3", " ", "r", " ", "\[Nu]"]]]], RowBox[List["6", " ", "r"]]]]], "}"]]]], "}"]], ",", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", FractionBox["2", "3"], ")"]], RowBox[List["2", "/", "3"]]]]], " ", "a", " ", SuperscriptBox["z", "r"]]], ",", FractionBox["1", "3"]]], "]"]]]]]], ")"]]]], ")"]]]]]]]]

MathML Form

<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> <msup> <mi> Bi </mi> <mo> ′ </mo> </msup> <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> <mi> r </mi> </mfrac> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mn> 2 </mn> <mrow> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mo> - </mo> <mfrac> <mn> 5 </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> 5 </mn> <mn> 6 </mn> </mfrac> </mrow> </msup> <mo> ⁢ </mo> <msqrt> <mi> π </mi> </msqrt> <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> / </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> 4 </mn> <mo> , </mo> <mn> 6 </mn> </mrow> <mrow> <mn> 2 </mn> <mo> , </mo> <mn> 3 </mn> </mrow> </msubsup> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mrow> <mo> - </mo> <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> </mrow> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> r </mi> </msup> </mrow> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 3 </mn> </mfrac> </mrow> <mo> ❘ </mo> <mtable> <mtr> <mtd> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 6 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> ν </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 6 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 5 </mn> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> ν </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mi> ν </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> r </mi> </mrow> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 3 </mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn> 0 </mn> <mo> , </mo> <mfrac> <mn> 2 </mn> <mn> 3 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 3 </mn> </mfrac> <mo> , </mo> <mrow> <mfrac> <mn> 2 </mn> <mn> 3 </mn> </mfrac> <mo> - </mo> <mi> ν </mi> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> r </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <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["4", ",", "6"]], RowBox[List["2", ",", "3"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List[TagBox[RowBox[List[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[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List["2", "-", RowBox[List["3", " ", "\[Nu]"]]]], ")"]]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List["5", "-", RowBox[List["3", " ", "\[Nu]"]]]], ")"]]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], "-", FractionBox["1", RowBox[List["3", " ", "r"]]], "+", "1"]], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "3"], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["2", "3"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "3"], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["2", "3"], "-", "\[Nu]"]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["-", "\[Nu]"]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["3", " ", "r", " ", "\[Nu]"]], "+", "2"]], RowBox[List["6", " ", "r"]]]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] </annotation> </semantics> </mrow> <mo> - </mo> <mrow> <msup> <mn> 9 </mn> <mi> ν </mi> </msup> <mo> ⁢ </mo> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mn> 4 </mn> <mo> , </mo> <mn> 6 </mn> </mrow> <mrow> <mn> 2 </mn> <mo> , </mo> <mn> 3 </mn> </mrow> </msubsup> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mrow> <mo> - </mo> <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> </mrow> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> r </mi> </msup> </mrow> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 3 </mn> </mfrac> </mrow> <mo> ❘ </mo> <mtable> <mtr> <mtd> <mrow> <mrow> <mfrac> <mi> ν </mi> <mn> 2 </mn> </mfrac> <mo> - </mo> <mfrac> <mn> 1 </mn> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> r </mi> </mrow> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 6 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 6 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> + </mo> <mn> 5 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 3 </mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn> 0 </mn> <mo> , </mo> <mfrac> <mn> 2 </mn> <mn> 3 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 3 </mn> </mfrac> <mo> , </mo> <mi> ν </mi> <mo> , </mo> <mrow> <mi> ν </mi> <mo> + </mo> <mfrac> <mn> 2 </mn> <mn> 3 </mn> </mfrac> </mrow> <mo> , </mo> <mfrac> <mrow> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> r </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mrow> <mn> 6 </mn> <mo> ⁢ </mo> <mi> r </mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["4", ",", "6"]], RowBox[List["2", ",", "3"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List[TagBox[RowBox[List[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[FractionBox["\[Nu]", "2"], "-", FractionBox["1", RowBox[List["3", " ", "r"]]], "+", "1"]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "\[Nu]"]], "+", "2"]], ")"]]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "\[Nu]"]], "+", "5"]], ")"]]]], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "3"], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["2", "3"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "3"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Nu]", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Nu]", "+", FractionBox["2", "3"]]], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[RowBox[List["3", " ", "r", " ", "\[Nu]"]], "-", "2"]], RowBox[List["6", " ", "r"]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <ci> BesselK </ci> <ci> ν </ci> <apply> <times /> <cn type='rational'> 2 <sep /> 3 </cn> <apply> <power /> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <ci> r </ci> </apply> </apply> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <ci> AiryBiPrime </ci> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <ci> r </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> r </ci> <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> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 5 <sep /> 3 </cn> </apply> </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'> 5 <sep /> 6 </cn> </apply> </apply> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> <apply> <power /> <apply> <power /> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <ci> r </ci> </apply> </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> <apply> <power /> <ci> z </ci> <ci> r </ci> </apply> </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 /> <cn type='rational'> 1 <sep /> 6 </cn> <apply> <plus /> <cn type='integer'> 2 </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'> 5 </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> <cn type='rational'> 1 <sep /> 3 </cn> </list> </list> <list> <list> <cn type='integer'> 0 </cn> <cn type='rational'> 2 <sep /> 3 </cn> </list> <list> <cn type='rational'> 1 <sep /> 3 </cn> <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> <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> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <cn type='rational'> 2 <sep /> 3 </cn> <cn type='rational'> 2 <sep /> 3 </cn> </apply> </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'> 2 </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'> 5 </cn> </apply> </apply> </list> <list> <cn type='rational'> 1 <sep /> 3 </cn> </list> </list> <list> <list> <cn type='integer'> 0 </cn> <cn type='rational'> 2 <sep /> 3 </cn> </list> <list> <cn type='rational'> 1 <sep /> 3 </cn> <ci> ν </ci> <apply> <plus /> <ci> ν </ci> <cn type='rational'> 2 <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> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <cn type='rational'> 2 <sep /> 3 </cn> <cn type='rational'> 2 <sep /> 3 </cn> </apply> </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>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[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["AiryBiPrime", "[", RowBox[List["a_", " ", SuperscriptBox["z_", "r_"]]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", FractionBox["5", "3"]]], "-", "\[Nu]"]]], " ", SuperscriptBox["3", RowBox[List[RowBox[List["-", FractionBox["5", "6"]]], "-", "\[Nu]"]]], " ", SqrtBox["\[Pi]"], " ", "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["2", "-", RowBox[List["3", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List["5", "-", RowBox[List["3", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List["1", "-", FractionBox["1", RowBox[List["3", " ", "r"]]], "-", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", FractionBox["1", "3"], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["2", "3"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "3"], ",", RowBox[List[FractionBox["2", "3"], "-", "\[Nu]"]], ",", RowBox[List["-", "\[Nu]"]], ",", RowBox[List["-", FractionBox[RowBox[List["2", "+", RowBox[List["3", " ", "r", " ", "\[Nu]"]]]], RowBox[List["6", " ", "r"]]]]]]], "}"]]]], "}"]], ",", RowBox[List[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["2", "+", RowBox[List["3", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List["5", "+", RowBox[List["3", " ", "\[Nu]"]]]], ")"]]]]]], "}"]], ",", RowBox[List["{", FractionBox["1", "3"], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["2", "3"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "3"], ",", "\[Nu]", ",", RowBox[List[FractionBox["2", "3"], "+", "\[Nu]"]], ",", FractionBox[RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["3", " ", "r", " ", "\[Nu]"]]]], RowBox[List["6", " ", "r"]]]]], "}"]]]], "}"]], ",", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", FractionBox["2", "3"], ")"]], RowBox[List["2", "/", "3"]]]]], " ", "a", " ", SuperscriptBox["z", "r"]]], ",", FractionBox["1", "3"]]], "]"]]]]]], ")"]]]], "r"]]]]]]]

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

2001-10-29