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http://functions.wolfram.com/03.03.21.0101.01
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Integrate[(1/z) ((a BesselJ[\[Mu], z] + b BesselY[\[Mu], z])
(Subscript[a, 1] BesselJ[\[Nu], z] + Subscript[b, 1] BesselY[\[Nu], z])),
z] == (1/((\[Mu] - \[Nu]) (\[Mu] + \[Nu])))
(((-a) z BesselJ[1 + \[Mu], z] BesselJ[\[Nu], z] +
a BesselJ[\[Mu], z] ((\[Mu] - \[Nu]) BesselJ[\[Nu], z] +
z BesselJ[1 + \[Nu], z]) +
b (z BesselJ[1 + \[Nu], z] BesselY[\[Mu], z] + BesselJ[\[Nu], z]
((\[Mu] - \[Nu]) BesselY[\[Mu], z] - z BesselY[1 + \[Mu], z])))
Subscript[a, 1] + ((-a) z BesselJ[1 + \[Mu], z] BesselY[\[Nu], z] +
a BesselJ[\[Mu], z] ((\[Mu] - \[Nu]) BesselY[\[Nu], z] +
z BesselY[1 + \[Nu], z]) +
b ((-z) BesselY[1 + \[Mu], z] BesselY[\[Nu], z] +
BesselY[\[Mu], z] ((\[Mu] - \[Nu]) BesselY[\[Nu], z] +
z BesselY[1 + \[Nu], z]))) Subscript[b, 1])
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List[FractionBox["1", "z"], RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["a", " ", RowBox[List["BesselJ", "[", RowBox[List["\[Mu]", ",", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["BesselY", "[", RowBox[List["\[Mu]", ",", "z"]], "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["a", "1"], " ", RowBox[List["BesselJ", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "+", RowBox[List[SubscriptBox["b", "1"], " ", RowBox[List["BesselY", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]]]], ")"]]]], ")"]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List[RowBox[List["(", RowBox[List["\[Mu]", "-", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["\[Mu]", "+", "\[Nu]"]], ")"]]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "a"]], " ", "z", " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["1", "+", "\[Mu]"]], ",", "z"]], "]"]], " ", RowBox[List["BesselJ", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "+", RowBox[List["a", " ", RowBox[List["BesselJ", "[", RowBox[List["\[Mu]", ",", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["\[Mu]", "-", "\[Nu]"]], ")"]], " ", RowBox[List["BesselJ", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "+", RowBox[List["z", " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", "z"]], "]"]]]]]], ")"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["z", " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", "z"]], "]"]], " ", RowBox[List["BesselY", "[", RowBox[List["\[Mu]", ",", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["BesselJ", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["\[Mu]", "-", "\[Nu]"]], ")"]], " ", RowBox[List["BesselY", "[", RowBox[List["\[Mu]", ",", "z"]], "]"]]]], "-", RowBox[List["z", " ", RowBox[List["BesselY", "[", RowBox[List[RowBox[List["1", "+", "\[Mu]"]], ",", "z"]], "]"]]]]]], ")"]]]]]], ")"]]]]]], ")"]], " ", SubscriptBox["a", "1"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "a"]], " ", "z", " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["1", "+", "\[Mu]"]], ",", "z"]], "]"]], " ", RowBox[List["BesselY", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "+", RowBox[List["a", " ", RowBox[List["BesselJ", "[", RowBox[List["\[Mu]", ",", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["\[Mu]", "-", "\[Nu]"]], ")"]], " ", RowBox[List["BesselY", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "+", RowBox[List["z", " ", RowBox[List["BesselY", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", "z"]], "]"]]]]]], ")"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "z"]], " ", RowBox[List["BesselY", "[", RowBox[List[RowBox[List["1", "+", "\[Mu]"]], ",", "z"]], "]"]], " ", RowBox[List["BesselY", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["BesselY", "[", RowBox[List["\[Mu]", ",", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["\[Mu]", "-", "\[Nu]"]], ")"]], " ", RowBox[List["BesselY", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "+", RowBox[List["z", " ", RowBox[List["BesselY", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", "z"]], "]"]]]]]], ")"]]]]]], ")"]]]]]], ")"]], " ", SubscriptBox["b", "1"]]]]], ")"]]]]]]]]
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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> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <msub> <mi> J </mi> <mi> μ </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <msub> <mi> Y </mi> <mi> μ </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> ⁢ </mo> <mrow> <msub> <mi> J </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> ⁢ </mo> <mrow> <msub> <mi> Y </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> z </mi> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> μ </mi> <mo> - </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> μ </mi> <mo> + </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> ⁢ </mo> <mi> z </mi> <mo> ⁢ </mo> <mrow> <msub> <mi> J </mi> <mrow> <mi> μ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mi> J </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <msub> <mi> J </mi> <mi> μ </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> μ </mi> <mo> - </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mi> J </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> z </mi> <mo> ⁢ </mo> <mrow> <msub> <mi> J </mi> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> ⁢ </mo> <mrow> <msub> <mi> J </mi> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mi> Y </mi> <mi> μ </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <msub> <mi> J </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> μ </mi> <mo> - </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mi> Y </mi> <mi> μ </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> z </mi> <mo> ⁢ </mo> <mrow> <msub> <mi> Y </mi> <mrow> <mi> μ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> ⁢ </mo> <mi> z </mi> <mo> ⁢ </mo> <mrow> <msub> <mi> J </mi> <mrow> <mi> μ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mi> Y </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <msub> <mi> J </mi> <mi> μ </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> μ </mi> <mo> - </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mi> Y </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> z </mi> <mo> ⁢ </mo> <mrow> <msub> <mi> Y </mi> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <msub> <mi> Y </mi> <mi> μ </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> μ </mi> <mo> - </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mi> Y </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> z </mi> <mo> ⁢ </mo> <mrow> <msub> <mi> Y </mi> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> z </mi> <mo> ⁢ </mo> <mrow> <msub> <mi> Y </mi> <mrow> <mi> μ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mi> Y </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <ci> BesselJ </ci> <ci> μ </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <ci> BesselY </ci> <ci> μ </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> BesselJ </ci> <ci> ν </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> BesselY </ci> <ci> ν </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> μ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <apply> <plus /> <ci> μ </ci> <ci> ν </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <ci> z </ci> <apply> <ci> BesselJ </ci> <apply> <plus /> <ci> μ </ci> <cn type='integer'> 1 </cn> </apply> <ci> z </ci> </apply> <apply> <ci> BesselJ </ci> <ci> ν </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <ci> a </ci> <apply> <ci> BesselJ </ci> <ci> μ </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> μ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <apply> <ci> BesselJ </ci> <ci> ν </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <ci> z </ci> <apply> <ci> BesselJ </ci> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <plus /> <apply> <times /> <ci> z </ci> <apply> <ci> BesselJ </ci> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> <ci> z </ci> </apply> <apply> <ci> BesselY </ci> <ci> μ </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <ci> BesselJ </ci> <ci> ν </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> μ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <apply> <ci> BesselY </ci> <ci> μ </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> z </ci> <apply> <ci> BesselY </ci> <apply> <plus /> <ci> μ </ci> <cn type='integer'> 1 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <ci> z </ci> <apply> <ci> BesselJ </ci> <apply> <plus /> <ci> μ </ci> <cn type='integer'> 1 </cn> </apply> <ci> z </ci> </apply> <apply> <ci> BesselY </ci> <ci> ν </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <ci> a </ci> <apply> <ci> BesselJ </ci> <ci> μ </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> μ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <apply> <ci> BesselY </ci> <ci> ν </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <ci> z </ci> <apply> <ci> BesselY </ci> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <plus /> <apply> <times /> <apply> <ci> BesselY </ci> <ci> μ </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> μ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <apply> <ci> BesselY </ci> <ci> ν </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <ci> z </ci> <apply> <ci> BesselY </ci> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> z </ci> <apply> <ci> BesselY </ci> <apply> <plus /> <ci> μ </ci> <cn type='integer'> 1 </cn> </apply> <ci> z </ci> </apply> <apply> <ci> BesselY </ci> <ci> ν </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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