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http://functions.wolfram.com/03.03.21.0094.01
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Integrate[z^(\[Alpha] - 1) (a BesselJ[\[Nu], z] + b BesselY[\[Nu], z])^2,
z] ==
-((z^(\[Alpha] - 2 \[Nu]) (2^(1 + 2 \[Nu]) b z^(2 \[Nu])
(-\[Alpha]^2 + 4 \[Nu]^2) (a + b Cot[Pi \[Nu]]) Csc[Pi \[Nu]]
Gamma[1 - \[Nu]] Gamma[1 + \[Nu]] HypergeometricPFQ[{1/2, \[Alpha]/2},
{1 + \[Alpha]/2, 1 - \[Nu], 1 + \[Nu]}, -z^2] +
\[Alpha] (b^2 (16^\[Nu] \[Alpha] + 2^(1 + 4 \[Nu]) \[Nu])
Csc[Pi \[Nu]]^2 Gamma[1 + \[Nu]]^2 HypergeometricPFQ[
{1/2 - \[Nu], \[Alpha]/2 - \[Nu]}, {1 - 2 \[Nu], 1 - \[Nu],
1 + \[Alpha]/2 - \[Nu]}, -z^2] - z^(4 \[Nu]) (-\[Alpha] + 2 \[Nu])
(a + b Cot[Pi \[Nu]])^2 Gamma[1 - \[Nu]]^2 HypergeometricPFQ[
{1/2 + \[Nu], \[Alpha]/2 + \[Nu]}, {1 + \[Nu], 1 + \[Alpha]/2 +
\[Nu], 1 + 2 \[Nu]}, -z^2])))/4^\[Nu])/
((-\[Alpha]^3 + 4 \[Alpha] \[Nu]^2) Gamma[1 - \[Nu]]^2 Gamma[1 + \[Nu]]^2)
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "1"]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["a", " ", RowBox[List["BesselJ", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["BesselY", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]]]], ")"]], "2"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List[SuperscriptBox["4", RowBox[List["-", "\[Nu]"]]], " ", SuperscriptBox["z", RowBox[List["\[Alpha]", "-", RowBox[List["2", " ", "\[Nu]"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List["1", "+", RowBox[List["2", " ", "\[Nu]"]]]]], " ", "b", " ", SuperscriptBox["z", RowBox[List["2", " ", "\[Nu]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["\[Alpha]", "2"]]], "+", RowBox[List["4", " ", SuperscriptBox["\[Nu]", "2"]]]]], ")"]], " ", RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cot", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]]]], ")"]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", FractionBox["\[Alpha]", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", FractionBox["\[Alpha]", "2"]]], ",", RowBox[List["1", "-", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["z", "2"]]]]], "]"]]]], "+", RowBox[List["\[Alpha]", " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["b", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["16", "\[Nu]"], " ", "\[Alpha]"]], "+", RowBox[List[SuperscriptBox["2", RowBox[List["1", "+", RowBox[List["4", " ", "\[Nu]"]]]]], " ", "\[Nu]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], "2"], " ", SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], "2"], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ",", RowBox[List[FractionBox["\[Alpha]", "2"], "-", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", RowBox[List["2", " ", "\[Nu]"]]]], ",", RowBox[List["1", "-", "\[Nu]"]], ",", RowBox[List["1", "+", FractionBox["\[Alpha]", "2"], "-", "\[Nu]"]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["z", "2"]]]]], "]"]]]], "-", RowBox[List[SuperscriptBox["z", RowBox[List["4", " ", "\[Nu]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "\[Alpha]"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cot", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]]]], ")"]], "2"], " ", SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Nu]"]], "]"]], "2"], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", RowBox[List[FractionBox["\[Alpha]", "2"], "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List["1", "+", FractionBox["\[Alpha]", "2"], "+", "\[Nu]"]], ",", RowBox[List["1", "+", RowBox[List["2", " ", "\[Nu]"]]]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["z", "2"]]]]], "]"]]]]]], ")"]]]]]], ")"]]]], ")"]]]], "/", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["\[Alpha]", "3"]]], "+", RowBox[List["4", " ", "\[Alpha]", " ", SuperscriptBox["\[Nu]", "2"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Nu]"]], "]"]], "2"], " ", SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], "2"]]], ")"]]]]]]]]
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</mn> <mo> ⁢ </mo> <mi> ν </mi> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <mi> α </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mn> 16 </mn> <mi> ν </mi> </msup> <mo> ⁢ </mo> <mi> α </mi> </mrow> <mo> + </mo> <mrow> <msup> <mn> 2 </mn> <mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mi> ν </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> csc </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 3 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> 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<times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <cn type='integer'> 4 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> α </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> ν </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> ν </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <ci> b </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> ν </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> α </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <apply> <cot /> <apply> <times /> <pi /> <ci> ν </ci> </apply> </apply> </apply> </apply> <apply> <csc /> <apply> <times /> <pi /> <ci> ν </ci> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <ci> α </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </list> <list> <apply> <plus /> <apply> <times /> <ci> α </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> </list> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> ν </ci> </apply> </apply> </apply> <apply> <times /> <ci> α </ci> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> 16 </cn> <ci> ν </ci> </apply> <ci> α </ci> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> ν </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <ci> ν </ci> </apply> </apply> <apply> <power /> <apply> <csc /> <apply> <times /> <pi /> <ci> ν </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> α </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> </list> <list> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> ν </ci> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> α </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <cn type='integer'> 1 </cn> </apply> </list> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> 4 </cn> <ci> ν </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> ν </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <apply> <cot /> <apply> <times /> <pi /> <ci> ν </ci> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <plus /> <ci> ν </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <ci> α </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> ν </ci> </apply> </list> <list> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <times /> <ci> α </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> ν </ci> </apply> <cn type='integer'> 1 </cn> </apply> </list> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> α </ci> <apply> <power /> <ci> ν </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> α </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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){kind=small}
